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Limit theorems for point processes under geometric constraints (and topological crackle)

2017-05-01
Takashi Owada , Robert J. Adler
We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A 
 We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Čech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via nonhomogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered. The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological "crackle;" the continued presence of spurious homology in samples of topological structures, despite increased sample size.
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Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows

2014-11-12
Takashi Owada , Gennady Samorodnitsky
We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying LĂ©vy measures. The limit process is a 
 We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying LĂ©vy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments that coincides on a part of its parameter space with a previously described process. The normalizing sequence and the limiting process are determined by the ergodic-theoretical properties of the flow underlying the integral representation of the process. These properties can be interpreted as determining how long the memory of the stationary infinitely divisible process is. We also establish functional convergence, in a strong distributional sense, for conservative pointwise dual ergodic maps preserving an infinite measure.
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Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments

2015-05-27
Takashi Owada , Gennady Samorodnitsky
We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-stable process. The length of memory in the stable process is parameterized by 
 We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-stable process. The length of memory in the stable process is parameterized by a certain ergodic-theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fréchet process. It is a self-similar process with $\alpha$-Fréchet marginals, and it has stationary max-increments, a property which we introduce in this paper. The functional limit theorem is established in the space $D[0,\infty)$ equipped with the Skorohod $M_{1}$-topology; in certain special cases the topology can be strengthened to the Skorohod $J_{1}$-topology.
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Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

2018-08-28
Takashi Owada
We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic 
 We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic topology. It then turns out that the growth rate of the Betti numbers and the properties of the limiting processes all depend on the distance of the region of interest from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes. We also derive the limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology.
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Functional central limit theorem for subgraph counting processes

2017-01-01
Takashi Owada
The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The 
 The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.
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Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows

2017-07-01
Paul Jung , Takashi Owada , Gennady Samorodnitsky
We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its 
 We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its long memory which is quantified by a Mittag–Leffler process induced by an associated Harris chain, at the discrete-time level. Previous results in Owada and Samorodnitsky [Ann. Probab. 43 (2015) 240–285] dealt with positive dependence in the increment process, whereas this paper derives the functional limit theorems under negative dependence. The negative dependence is due to cancellations arising from Gaussian-type fluctuations of functionals of the associated Harris chain. The new types of limiting processes involve stable random measures, due to heavy tails, Mittag–Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations. Along the way, we prove a function central limit theorem for fluctuations of functionals of Harris chains which is of independent interest as it extends a result of Chen [Probab. Theory Related Fields 116 (2000) 89–123].
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A functional non-central limit theorem for multiple-stable processes with long-range dependence

2020-04-24
Shuyang Bai , Takashi Owada , Yizao Wang
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Functional limit theorems for the euler characteristic process in the critical regime

2021-03-01
Andrew M. Thomas , Takashi Owada
Abstract This study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $\mathbb{R}^d$ , and the connectivity radius 
 Abstract This study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $\mathbb{R}^d$ , and the connectivity radius governing the formation of simplices is taken as a function of the time parameter t , which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have nontrivial topology. We establish two ‘functional-level’ limit theorems, a strong law of large numbers and a central limit theorem, for the appropriately normalized Euler characteristic process.
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Limit Theory for the Sample Autocovariance for Heavy-Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows

2014-07-02
Takashi Owada
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Topological crackle of heavy-tailed moving average processes

2019-01-11
Takashi Owada

Sub-tree counts on hyperbolic random geometric graphs

2022-06-13
Takashi Owada , D. Yogeshwaran
Abstract The hyperbolic random geometric graph was introduced by Krioukov et al. ( Phys. Rev. E 82 , 2010). Among many equivalent models for the hyperbolic space, we study the 
 Abstract The hyperbolic random geometric graph was introduced by Krioukov et al. ( Phys. Rev. E 82 , 2010). Among many equivalent models for the hyperbolic space, we study the d -dimensional PoincarĂ© ball ( $d\ge 2$ ), with a general connectivity radius. While many phase transitions are known for the expectation asymptotics of certain subgraph counts, very little is known about the second-order results. Two of the distinguishing characteristics of geometric graphs on the hyperbolic space are the presence of tree-like hierarchical structures and the power-law behaviour of the degree distribution. We aim to reveal such characteristics in detail by investigating the behaviour of sub-tree counts. We show multiple phase transitions for expectation and variance in the resulting hyperbolic geometric graph. In particular, the expectation and variance of the sub-tree counts exhibit an intricate dependence on the degree sequence of the tree under consideration. Additionally, unlike the thermodynamic regime of the Euclidean random geometric graph, the expectation and variance may exhibit different growth rates, which is indicative of power-law behaviour. Finally, we also prove a normal approximation for sub-tree counts using the Malliavin–Stein method of Last et al. ( Prob. Theory Relat. Fields 165 , 2016), along with the Palm calculus for Poisson point processes.
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Convergence of persistence diagrams for topological crackle

2020-04-27
Takashi Owada , Omer Bobrowski
In this paper, we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the 
 In this paper, we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of topological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to topological cycles generated by random points far away from the bulk of other points, when the support is unbounded. We establish weak convergence results for persistence diagrams – a point process representation for persistent homology, where each topological cycle is represented by its $({\mathit{birth},\mathit{death}})$ coordinates. In this work, we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. Using this framework, we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a shorter lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.
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Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex

2021-04-22
Takashi Owada , Gennady Samorodnitsky , Gugan Thoppe
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Limit Theorems for Point Processes under Geometric Constraints (and Topological Crackle)

2015-01-01
Takashi Owada , Robert J. Adler
We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A 
 We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of \v{C}ech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via non-homogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered. The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological `crackle'; the continued presence of spurious homology in samples of topological structures, despite increased sample size.
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Limit Theorems for the Sum of Persistence Barcodes

2016-01-01
Takashi Owada
Topological Data Analysis (TDA) refers to an approach that uses concepts from algebraic topology to study the "shapes" of datasets. The main focus of this paper is persistent homology, a 
 Topological Data Analysis (TDA) refers to an approach that uses concepts from algebraic topology to study the "shapes" of datasets. The main focus of this paper is persistent homology, a ubiquitous tool in TDA. Basing our study on this, we investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb R^d$. In particular, we establish various limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology. It then turns out that the growth rate of the sum of the bar lengths and the properties of the limiting processes all depend on the distance of the region of interest in $\mathbb R^d$ from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes.

Limit theorems for process-level Betti numbers for sparse, critical, and Poisson regimes

2018-01-01
Takashi Owada , Andrew M. Thomas
The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the $d$-dimensional Euclidean 
 The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the $d$-dimensional Euclidean space $\mathbb{R}^d$. We consider the case where the points of the Čech complex are generated by a Poisson process with intensity $nf$ for a probability density $f$. We look at the cases where the behavior of the connectivity radius of Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse and Poisson regimes, as well when the connectivity radius is on the order of $n^{-1/d}$, the critical regime. We establish limit theorems in all of the aforementioned regimes, a central limit theorem for the sparse and critical regimes, and a Poisson limit theorem for the Poisson regime. When the connectivity radius of the Čech complex is $o(n^{-1/d})$, i.e., the sparse and Poisson regimes, we can decompose the limiting processes into a time-changed Brownian motion and a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.

Large deviation principle for geometric and topological functionals and associated point processes

2023-10-01
Christian Hirsch , Takashi Owada
We prove a large deviation principle for the point process associated to k-element connected components in Rd with respect to the connectivity radii rn→∞. The random points are generated from 
 We prove a large deviation principle for the point process associated to k-element connected components in Rd with respect to the connectivity radii rn→∞. The random points are generated from a homogeneous Poisson point process or the corresponding binomial point process, so that (rn)n≄1 satisfies nkrnd(k−1)→∞ and nrnd→0 as n→∞ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.
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Functional central limit theorem for negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows

2015-01-01
Paul Jung , Takashi Owada , Gennady Samorodnitsky
We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only 
 We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only positive dependence could be treated. The negative dependence involves cancellations of the Gaussian second order. This leads to new types of limiting processes involving stable random measures, due to heavy tails, Mittag-Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations.

Convergence of Persistence Diagrams for Topological Crackle

2018-01-01
Takashi Owada , Omer Bobrowski
In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the 
 In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of homological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to homological cycles generated by "noisy" samples where the support is unbounded. We aim to establish weak convergence results for persistence diagrams - a point process representation for persistent homology, where each homological cycle is represented by its (birth,death) coordinates. In this work we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. In this framework we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a short lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.
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Functional limit theorems for the Euler characteristic process in the critical regime

2019-01-01
Andrew M. Thomas , Takashi Owada
This study presents functional limit theorems for the Euler characteristic of Vietoris-Rips complexes. The points are drawn from a non-homogeneous Poisson process on $\mathbb{R}^d$, and the connectivity radius governing the 
 This study presents functional limit theorems for the Euler characteristic of Vietoris-Rips complexes. The points are drawn from a non-homogeneous Poisson process on $\mathbb{R}^d$, and the connectivity radius governing the formation of simplices is taken as a function of time parameter $t$, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have non-trivial topology. We establish two "functional-level" limit theorems, a strong law of large numbers and a central limit theorem for the appropriately normalized Euler characteristic process.
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Functional strong law of large numbers for Betti numbers in the tail

2022-05-31
Takashi Owada , Zifu Wei
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Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

2021-05-20
Andrew M. Thomas , Takashi Owada
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Convergence of persistence diagram in the sparse regime

2022-12-01
Takashi Owada
The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic 
 The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Čech filtration over a scaled random sample rn−1Xn={rn−1X1,
,rn−1Xn}, such that rn→0 as n→∞. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: nrnd→0, n→∞. In this setting, we show that the asymptotics of the kth persistence diagram depends on the limit value of the sequence nk+2rnd(k+1). If nk+2rnd(k+1)→∞, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If rn decays faster so that nk+2rnd(k+1)→c∈(0,∞), the persistence diagram weakly converges to a limiting point process without normalization. Finally, if nk+2rnd(k+1)→0, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the M0-topology.
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Large deviations for the volume of k-nearest neighbor balls

2023-01-01
Christian Hirsch , Taegyu Kang , Takashi Owada
This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a 
 This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of k-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of M0-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most k in a random geometric graph in the dense regime.
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Limit theory for <i>U</i>-statistics under geometric and topological constraints with rare events

2022-09-14
Takashi Owada
Abstract We study the geometric and topological features of U -statistics of order k when the k -tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, 
 Abstract We study the geometric and topological features of U -statistics of order k when the k -tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, we establish the convergence of U -statistics in vague topology, while the structure of a non-degenerate limit measure is also revealed. Our general result shows various limit theorems for geometric and topological statistics, including persistent Betti numbers of Čech complexes, the volume of simplices, a functional of the Morse critical points, and values of the min-type distance function. The required vague convergence can be obtained as a result of the limit theorem for point processes induced by U -statistics. The latter convergence particularly occurs in the $\mathcal M_0$ -topology.

Large deviations for subcomplex counts and Betti numbers in multiparameter simplicial complexes

2023-02-16
Gennady Samorodnitsky , Takashi Owada
We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical ErdƑs–RĂ©nyi graph. We investigate appearance of “unusual” topological structures in the complex from the point 
 We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical ErdƑs–RĂ©nyi graph. We investigate appearance of “unusual” topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices of the multiparameter simplicial complexes. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.
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Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

2020-01-01
Andrew M. Thomas , Takashi Owada
To recover the topology of a manifold in the presence of heavy tailed or exponentially decaying noise, one must understand the behavior of geometric complexes whose points lie in the 
 To recover the topology of a manifold in the presence of heavy tailed or exponentially decaying noise, one must understand the behavior of geometric complexes whose points lie in the tail of these noise distributions. This study advances this line of inquiry, and demonstrates functional strong laws of large numbers for the Euler characteristic process of random geometric complexes formed by random points outside of an expanding ball in $\mathbb{R}^d$. When the points are drawn from a heavy tailed distribution with a regularly varying tail, the Euler characteristic process grows at a regularly varying rate, and the scaled process converges uniformly and almost surely to a smooth function. When the points are drawn from a distribution with an exponentially decaying tail, the Euler characteristic process grows logarithmically, and the scaled process converges to another smooth function in the same sense. All of the limit theorems take place when the points inside the expanding ball are densely distributed, so that the simplex counts outside of the ball of all dimensions contribute to the Euler characteristic process.

Sub-tree counts on hyperbolic random geometric graphs

2018-01-01
Takashi Owada , D. Yogeshwaran
We study the hyperbolic random geometric graph introduced in Krioukov et al. For a sequence $R_n \to \infty$, we define these graphs to have the vertex set as Poisson points 
 We study the hyperbolic random geometric graph introduced in Krioukov et al. For a sequence $R_n \to \infty$, we define these graphs to have the vertex set as Poisson points distributed uniformly in balls $B(0,R_n) \subset B_d^{\alpha}$, the $d$-dimensional Poincar\'e ball (unit d-ball with the Poincar\'e metric $d_{\alpha}$ corresponding to negative curvature $-\alpha^2, \alpha > 0$) by connecting any two points within a distance $R_n$ according to the metric $d_{\zeta}, \zeta > 0$. Denoting these graphs by $HG_n(R_n ; \alpha, \zeta)$, we study asymptotic counts of copies of a fixed tree $\Gamma_k$ (with the ordered degree sequence $d_{(1)} \leq \ldots \leq d_{(k)}$) in $HG_n(R_n ; \alpha, \zeta)$. Unlike earlier works, we count more involved structures, allowing for $d > 2$, and in many places, more general choices of $R_n$ rather than $R_n = 2[\zeta (d-1)]^{-1}\log (n/ \nu), \nu \in (0,\infty)$. The latter choice of $R_n$ for $\alpha / \zeta > 1/2$ corresponds to the thermodynamic regime. We show multiple phase transitions in $HG_n(R_n ; \alpha, \zeta)$ as $\alpha / \zeta$ increases, i.e., the space $B_d^{\alpha}$ becomes more hyperbolic. In particular, our analyses reveal that the sub-tree counts exhibit an intricate dependence on the degree sequence $d_{(1)},\ldots,d_{(k)}$ of $\Gamma_k$ as well as the ratio $\alpha/\zeta$. Under a more general radius regime $R_n$ than that described above, we investigate the asymptotics of the expectation and variance of sub-tree counts. Moreover, we prove the corresponding central limit theorem as well. Our proofs rely crucially on a careful analysis of the sub-tree counts near the boundary using Palm calculus for Poisson point processes along with estimates for the hyperbolic metric and measure. For the central limit theorem, we use the abstract normal approximation result from Last et al. derived using the Malliavin-Stein method.
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Large deviations for hyperbolic $k$-nearest neighbor balls

2023-01-01
Christian Hirsch , Moritz Otto , Takashi Owada +1 more
We prove a large deviation principle for the point process of large Poisson $k$-nearest neighbor balls in hyperbolic space. More precisely, we consider a stationary Poisson point process of unit 
 We prove a large deviation principle for the point process of large Poisson $k$-nearest neighbor balls in hyperbolic space. More precisely, we consider a stationary Poisson point process of unit intensity in a growing sampling window in hyperbolic space. We further take a growing sequence of thresholds such that there is a diverging expected number of Poisson points whose $k$-nearest neighbor ball has a volume exceeding this threshold. Then, the point process of exceedances satisfies a large deviation principle whose rate function is described in terms of a relative entropy. The proof relies on a fine coarse-graining technique such that inside the resulting blocks the exceedances are approximated by independent Poisson point processes.
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Functional Central Limit Theorem for Subgraph Counting Processes

2015-01-01
Takashi Owada
The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific 
 The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.
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A functional non-central limit theorem for multiple-stable processes with long-range dependence

2019-01-01
Shuyang Bai , Takashi Owada , Yizao Wang
A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using 
 A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets. The joint moments of the local times are computed, which play the key in the proof and are also of independent interest.
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Limit Theory for the Sample Autocovariance for Heavy Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows

2013-01-01
Takashi Owada
This study aims to develop the limit theorems on the sample autocovariances and sample autocorrelations for certain stationary infinitely divisible processes. We consider the case where the infinitely divisible process 
 This study aims to develop the limit theorems on the sample autocovariances and sample autocorrelations for certain stationary infinitely divisible processes. We consider the case where the infinitely divisible process has heavy tail marginals and is generated by a conservative flow. Interestingly, the growth rate of the sample autocovariances is determined by not only heavy tailedness of the marginals but also memory length of the process. Although this feature was first observed by \cite{resnick:samorodnitsky:xue:2000} for some very specific processes, we will propose a more general framework from the viewpoint of infinite ergodic theory. Consequently, the asymptotics of the sample autocovariances can be more comprehensively discussed.

Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex

2020-01-01
Takashi Owada , Gennady Samorodnitsky , Gugan Thoppe
Topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since the randomness there is usually 
 Topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since the randomness there is usually governed by a single parameter. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex and, more importantly, of its dynamic analogue that we introduce here. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. The dynamic versions of the clique complex and the Linial-Meshulum complex are special cases of our setup. Our key result concerns the regime where face-counts of a particular dimension dominate. We show that the Betti numbers corresponding to this dimension and the Euler characteristic satisfy functional strong law of large numbers and functional central limit theorems. Surprisingly, in the latter result, the limiting Gaussian process depends only upon the dynamics in the smallest non-trivial dimension.
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Empirical Likelihood Estimation of Levy Processes (Revised in March 2005)

2004-01-01
Naoto Kunitomo , Takashi Owada
We propose a new parameter estimation procedure for the Levy processes and the class of infinitely divisible distribution. We shall show that the empirical likelihood method gives an easy way 
 We propose a new parameter estimation procedure for the Levy processes and the class of infinitely divisible distribution. We shall show that the empirical likelihood method gives an easy way to estimate the key parameters of the infinitely divisible distributions including the class of stable distributions as a special case. The maximum empirical likelihood estimator by using the empirical characteristic functions gives the consistency, the asymptotic normality, and the asymptotic efficiency for the key parameters when the number of restrictions on the empirical characteristic functions is large. Test procedures can be also developed. Some extensions to the estimating equations problem with the infinitely divisible distributions are discussed.

Convergence of persistence diagram in the subcritical regime

2021-03-24
Takashi Owada
The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Cech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic 
 The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Cech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Cech filtration over a scaled random sample $r_n^{-1}\mathcal X_n = \{ r_n^{-1}X_1,\dots, r_n^{-1}X_n \}$, such that $r_n\to 0$ as $n\to\infty$. We treat persistence diagrams as a point process and establish their limit theorems in the subcritical regime: $nr_n^d\to0$, $n\to\infty$. In this setting, we show that the asymptotics of the $k$th persistence diagram depends on the limit value of the sequence $n^{k+2}r_n^{d(k+1)}$. If $n^{k+2}r_n^{d(k+1)} \to \infty$, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If $r_n$ decays faster so that $n^{k+2}r_n^{d(k+1)} \to c\in (0,\infty)$, the persistence diagram weakly converges to a limiting point process without normalization. Finally, if $n^{k+2}r_n^{d(k+1)} \to 0$, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the $\mathcal M_0$-topology.

Convergence of persistence diagram in the sparse regime

2021-01-01
Takashi Owada
The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with \v{C}ech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic 
 The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with \v{C}ech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider \v{C}ech filtration over a scaled random sample $r_n^{-1}\mathcal X_n = \{ r_n^{-1}X_1,\dots, r_n^{-1}X_n \}$, such that $r_n\to 0$ as $n\to\infty$. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: $nr_n^d\to0$, $n\to\infty$. In this setting, we show that the asymptotics of the $k$th persistence diagram depends on the limit value of the sequence $n^{k+2}r_n^{d(k+1)}$. If $n^{k+2}r_n^{d(k+1)} \to \infty$, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If $r_n$ decays faster so that $n^{k+2}r_n^{d(k+1)} \to c\in (0,\infty)$, the persistence diagram weakly converges to a limiting point process without normalization. Finally, if $n^{k+2}r_n^{d(k+1)} \to 0$, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the $\mathcal M_0$-topology.
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Large deviation principle for geometric and topological functionals and associated point processes

2022-01-01
Christian Hirsch , Takashi Owada
We prove a large deviation principle for the point process associated to $k$-element connected components in $\mathbb R^d$ with respect to the connectivity radii $r_n\to\infty$. The random points are generated 
 We prove a large deviation principle for the point process associated to $k$-element connected components in $\mathbb R^d$ with respect to the connectivity radii $r_n\to\infty$. The random points are generated from a homogeneous Poisson point process, so that $(r_n)_{n\ge1}$ satisfies $n^kr_n^{d(k-1)}\to\infty$ and $nr_n^d\to0$ as $n\to\infty$ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.
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Large deviations for subcomplex counts and Betti numbers in multi-parameter simplicial complexes

2022-01-01
Gennady Samorodnitsky , Takashi Owada
We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erd\"os-R\'enyi graph. We investigate appearance of "unusual" topological structures in the complex from the point 
 We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erd\"os-R\'enyi graph. We investigate appearance of "unusual" topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices at the critical dimension and below. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.

Large deviations for the volume of $k$-nearest neighbor balls

2022-01-01
Christian Hirsch , Taegyu Kang , Takashi Owada
This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a 
 This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $\mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most $k$ in a geometric graph in the dense regime.
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Functional strong law of large numbers for Betti numbers in the tail

2021-01-01
Takashi Owada , Zifu Wei
The objective of this paper is to investigate the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers 
 The objective of this paper is to investigate the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers for Betti numbers, a basic quantifier of algebraic topology, of a geometric complex outside an open ball of radius $R_n$, such that $R_n\to\infty$ as the sample size $n$ increases. The nature of the obtained law of large numbers is determined by the decay rate of a probability density. It especially depends on whether the tail of a density decays at a regularly varying rate or an exponentially decaying rate. The nature of the limit theorem depends also on how rapidly $R_n$ diverges. In particular, if $R_n$ diverges sufficiently slowly, the limiting function in the law of large numbers is crucially affected by the emergence of arbitrarily large connected components supporting topological cycles in the limit.
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Limit theorems for critical faces above the vanishing threshold

2023-01-01
Zifu Wei , Takashi Owada , D. Yogeshwaran
We investigate convergence of point processes associated with critical faces for a \v{C}ech filtration built over a homogeneous Poisson point process in the $d$-dimensional flat torus. The convergence of our 
 We investigate convergence of point processes associated with critical faces for a \v{C}ech filtration built over a homogeneous Poisson point process in the $d$-dimensional flat torus. The convergence of our point process is established in terms of the $\mathcal M_0$-topology, when the connecting radius of a \v{C}ech complex decays to $0$, so slowly that critical faces are even less likely to occur than those in the regime of threshold for homological connectivity. We also obtain a series of limit theorems for positive and negative critical faces, all of which are considerably analogous to those for critical faces.

Limit theorems under heavy-tailed scenario in the age dependent random connection models

2024-09-08
Christian Hirsch , Takashi Owada
This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random 
 This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable assumptions on the shape of trees. The proof relies on an intermediate result on weak convergence of associated point processes towards a Poisson point process. Additionally, we prove the same type of results for the clique counts. Here, a crucial ingredient includes the expectation asymptotics for clique counts, which itself is a result of independent interest.
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Degree counts in random simplicial complexes of the preferential attachment type

2024-10-22
Takashi Owada , Gennady Samorodnitsky
We extend the classical preferential attachment random graph model to random simplicial complexes. At each stage of the model, we choose one of the existing $k$-simplices with probability proportional to 
 We extend the classical preferential attachment random graph model to random simplicial complexes. At each stage of the model, we choose one of the existing $k$-simplices with probability proportional to its $k$-degree. The chosen $k$-simplex then forms a $(k+1)$-simplex with a newly arriving vertex. We establish a strong law of large numbers for the degree counts across multiple dimensions. The limiting probability mass function is expressed as a mixture of mass functions of different types of negative binomial random variables. This limiting distribution has power-law characteristics and we explore the limiting extremal dependence of the degree counts across different dimensions in the framework of multivariate regular variation. Finally, we prove multivariate weak convergence, under appropriate normalization, of degree counts in different dimensions, of ordered $k$-simplices. The resulting weak limit can be represented as a function of independent linear birth processes with immigration.
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Limit theorems for critical faces above the vanishing threshold

2025-01-01
Zifu Wei , Takashi Owada , D. Yogeshwaran

Limit theorems for high-dimensional Betti numbers in the multiparameter random simplicial complexes

2025-03-01
Takashi Owada , Gennady Samorodnitsky

Large deviations for the volume of hyperbolic k-nearest neighbor balls

2025-05-01
Christian Hirsch , Moritz Otto , Takashi Owada +1 more

Large deviations for the volume of hyperbolic k-nearest neighbor balls

2025-05-01
Christian Hirsch , Moritz Otto , Takashi Owada +1 more

Limit theorems for high-dimensional Betti numbers in the multiparameter random simplicial complexes

2025-03-01
Takashi Owada , Gennady Samorodnitsky

Limit theorems for critical faces above the vanishing threshold

2025-01-01
Zifu Wei , Takashi Owada , D. Yogeshwaran

Degree counts in random simplicial complexes of the preferential attachment type

2024-10-22
Takashi Owada , Gennady Samorodnitsky
We extend the classical preferential attachment random graph model to random simplicial complexes. At each stage of the model, we choose one of the existing $k$-simplices with probability proportional to 
 We extend the classical preferential attachment random graph model to random simplicial complexes. At each stage of the model, we choose one of the existing $k$-simplices with probability proportional to its $k$-degree. The chosen $k$-simplex then forms a $(k+1)$-simplex with a newly arriving vertex. We establish a strong law of large numbers for the degree counts across multiple dimensions. The limiting probability mass function is expressed as a mixture of mass functions of different types of negative binomial random variables. This limiting distribution has power-law characteristics and we explore the limiting extremal dependence of the degree counts across different dimensions in the framework of multivariate regular variation. Finally, we prove multivariate weak convergence, under appropriate normalization, of degree counts in different dimensions, of ordered $k$-simplices. The resulting weak limit can be represented as a function of independent linear birth processes with immigration.
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Limit theorems under heavy-tailed scenario in the age dependent random connection models

2024-09-08
Christian Hirsch , Takashi Owada
This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random 
 This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable assumptions on the shape of trees. The proof relies on an intermediate result on weak convergence of associated point processes towards a Poisson point process. Additionally, we prove the same type of results for the clique counts. Here, a crucial ingredient includes the expectation asymptotics for clique counts, which itself is a result of independent interest.
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Large deviation principle for geometric and topological functionals and associated point processes

2023-10-01
Christian Hirsch , Takashi Owada
We prove a large deviation principle for the point process associated to k-element connected components in Rd with respect to the connectivity radii rn→∞. The random points are generated from 
 We prove a large deviation principle for the point process associated to k-element connected components in Rd with respect to the connectivity radii rn→∞. The random points are generated from a homogeneous Poisson point process or the corresponding binomial point process, so that (rn)n≄1 satisfies nkrnd(k−1)→∞ and nrnd→0 as n→∞ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.
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Large deviations for subcomplex counts and Betti numbers in multiparameter simplicial complexes

2023-02-16
Gennady Samorodnitsky , Takashi Owada
We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical ErdƑs–RĂ©nyi graph. We investigate appearance of “unusual” topological structures in the complex from the point 
 We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical ErdƑs–RĂ©nyi graph. We investigate appearance of “unusual” topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices of the multiparameter simplicial complexes. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.
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Large deviations for hyperbolic $k$-nearest neighbor balls

2023-01-01
Christian Hirsch , Moritz Otto , Takashi Owada +1 more
We prove a large deviation principle for the point process of large Poisson $k$-nearest neighbor balls in hyperbolic space. More precisely, we consider a stationary Poisson point process of unit 
 We prove a large deviation principle for the point process of large Poisson $k$-nearest neighbor balls in hyperbolic space. More precisely, we consider a stationary Poisson point process of unit intensity in a growing sampling window in hyperbolic space. We further take a growing sequence of thresholds such that there is a diverging expected number of Poisson points whose $k$-nearest neighbor ball has a volume exceeding this threshold. Then, the point process of exceedances satisfies a large deviation principle whose rate function is described in terms of a relative entropy. The proof relies on a fine coarse-graining technique such that inside the resulting blocks the exceedances are approximated by independent Poisson point processes.
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Large deviations for the volume of k-nearest neighbor balls

2023-01-01
Christian Hirsch , Taegyu Kang , Takashi Owada
This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a 
 This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of k-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of M0-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most k in a random geometric graph in the dense regime.
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Limit theorems for critical faces above the vanishing threshold

2023-01-01
Zifu Wei , Takashi Owada , D. Yogeshwaran
We investigate convergence of point processes associated with critical faces for a \v{C}ech filtration built over a homogeneous Poisson point process in the $d$-dimensional flat torus. The convergence of our 
 We investigate convergence of point processes associated with critical faces for a \v{C}ech filtration built over a homogeneous Poisson point process in the $d$-dimensional flat torus. The convergence of our point process is established in terms of the $\mathcal M_0$-topology, when the connecting radius of a \v{C}ech complex decays to $0$, so slowly that critical faces are even less likely to occur than those in the regime of threshold for homological connectivity. We also obtain a series of limit theorems for positive and negative critical faces, all of which are considerably analogous to those for critical faces.

Convergence of persistence diagram in the sparse regime

2022-12-01
Takashi Owada
The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic 
 The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Čech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Čech filtration over a scaled random sample rn−1Xn={rn−1X1,
,rn−1Xn}, such that rn→0 as n→∞. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: nrnd→0, n→∞. In this setting, we show that the asymptotics of the kth persistence diagram depends on the limit value of the sequence nk+2rnd(k+1). If nk+2rnd(k+1)→∞, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If rn decays faster so that nk+2rnd(k+1)→c∈(0,∞), the persistence diagram weakly converges to a limiting point process without normalization. Finally, if nk+2rnd(k+1)→0, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the M0-topology.
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Limit theory for <i>U</i>-statistics under geometric and topological constraints with rare events

2022-09-14
Takashi Owada
Abstract We study the geometric and topological features of U -statistics of order k when the k -tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, 
 Abstract We study the geometric and topological features of U -statistics of order k when the k -tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, we establish the convergence of U -statistics in vague topology, while the structure of a non-degenerate limit measure is also revealed. Our general result shows various limit theorems for geometric and topological statistics, including persistent Betti numbers of Čech complexes, the volume of simplices, a functional of the Morse critical points, and values of the min-type distance function. The required vague convergence can be obtained as a result of the limit theorem for point processes induced by U -statistics. The latter convergence particularly occurs in the $\mathcal M_0$ -topology.

Sub-tree counts on hyperbolic random geometric graphs

2022-06-13
Takashi Owada , D. Yogeshwaran
Abstract The hyperbolic random geometric graph was introduced by Krioukov et al. ( Phys. Rev. E 82 , 2010). Among many equivalent models for the hyperbolic space, we study the 
 Abstract The hyperbolic random geometric graph was introduced by Krioukov et al. ( Phys. Rev. E 82 , 2010). Among many equivalent models for the hyperbolic space, we study the d -dimensional PoincarĂ© ball ( $d\ge 2$ ), with a general connectivity radius. While many phase transitions are known for the expectation asymptotics of certain subgraph counts, very little is known about the second-order results. Two of the distinguishing characteristics of geometric graphs on the hyperbolic space are the presence of tree-like hierarchical structures and the power-law behaviour of the degree distribution. We aim to reveal such characteristics in detail by investigating the behaviour of sub-tree counts. We show multiple phase transitions for expectation and variance in the resulting hyperbolic geometric graph. In particular, the expectation and variance of the sub-tree counts exhibit an intricate dependence on the degree sequence of the tree under consideration. Additionally, unlike the thermodynamic regime of the Euclidean random geometric graph, the expectation and variance may exhibit different growth rates, which is indicative of power-law behaviour. Finally, we also prove a normal approximation for sub-tree counts using the Malliavin–Stein method of Last et al. ( Prob. Theory Relat. Fields 165 , 2016), along with the Palm calculus for Poisson point processes.
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Functional strong law of large numbers for Betti numbers in the tail

2022-05-31
Takashi Owada , Zifu Wei
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Large deviation principle for geometric and topological functionals and associated point processes

2022-01-01
Christian Hirsch , Takashi Owada
We prove a large deviation principle for the point process associated to $k$-element connected components in $\mathbb R^d$ with respect to the connectivity radii $r_n\to\infty$. The random points are generated 
 We prove a large deviation principle for the point process associated to $k$-element connected components in $\mathbb R^d$ with respect to the connectivity radii $r_n\to\infty$. The random points are generated from a homogeneous Poisson point process, so that $(r_n)_{n\ge1}$ satisfies $n^kr_n^{d(k-1)}\to\infty$ and $nr_n^d\to0$ as $n\to\infty$ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.
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Large deviations for subcomplex counts and Betti numbers in multi-parameter simplicial complexes

2022-01-01
Gennady Samorodnitsky , Takashi Owada
We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erd\"os-R\'enyi graph. We investigate appearance of "unusual" topological structures in the complex from the point 
 We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erd\"os-R\'enyi graph. We investigate appearance of "unusual" topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices at the critical dimension and below. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.

Large deviations for the volume of $k$-nearest neighbor balls

2022-01-01
Christian Hirsch , Taegyu Kang , Takashi Owada
This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a 
 This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $\mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most $k$ in a geometric graph in the dense regime.
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Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

2021-05-20
Andrew M. Thomas , Takashi Owada
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Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex

2021-04-22
Takashi Owada , Gennady Samorodnitsky , Gugan Thoppe
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Convergence of persistence diagram in the subcritical regime

2021-03-24
Takashi Owada
The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Cech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic 
 The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with Cech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider Cech filtration over a scaled random sample $r_n^{-1}\mathcal X_n = \{ r_n^{-1}X_1,\dots, r_n^{-1}X_n \}$, such that $r_n\to 0$ as $n\to\infty$. We treat persistence diagrams as a point process and establish their limit theorems in the subcritical regime: $nr_n^d\to0$, $n\to\infty$. In this setting, we show that the asymptotics of the $k$th persistence diagram depends on the limit value of the sequence $n^{k+2}r_n^{d(k+1)}$. If $n^{k+2}r_n^{d(k+1)} \to \infty$, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If $r_n$ decays faster so that $n^{k+2}r_n^{d(k+1)} \to c\in (0,\infty)$, the persistence diagram weakly converges to a limiting point process without normalization. Finally, if $n^{k+2}r_n^{d(k+1)} \to 0$, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the $\mathcal M_0$-topology.

Functional limit theorems for the euler characteristic process in the critical regime

2021-03-01
Andrew M. Thomas , Takashi Owada
Abstract This study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $\mathbb{R}^d$ , and the connectivity radius 
 Abstract This study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $\mathbb{R}^d$ , and the connectivity radius governing the formation of simplices is taken as a function of the time parameter t , which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have nontrivial topology. We establish two ‘functional-level’ limit theorems, a strong law of large numbers and a central limit theorem, for the appropriately normalized Euler characteristic process.
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Convergence of persistence diagram in the sparse regime

2021-01-01
Takashi Owada
The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with \v{C}ech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic 
 The objective of this paper is to examine the asymptotic behavior of persistence diagrams associated with \v{C}ech filtration. A persistence diagram is a graphical descriptor of a topological and algebraic structure of geometric objects. We consider \v{C}ech filtration over a scaled random sample $r_n^{-1}\mathcal X_n = \{ r_n^{-1}X_1,\dots, r_n^{-1}X_n \}$, such that $r_n\to 0$ as $n\to\infty$. We treat persistence diagrams as a point process and establish their limit theorems in the sparse regime: $nr_n^d\to0$, $n\to\infty$. In this setting, we show that the asymptotics of the $k$th persistence diagram depends on the limit value of the sequence $n^{k+2}r_n^{d(k+1)}$. If $n^{k+2}r_n^{d(k+1)} \to \infty$, the scaled persistence diagram converges to a deterministic Radon measure almost surely in the vague metric. If $r_n$ decays faster so that $n^{k+2}r_n^{d(k+1)} \to c\in (0,\infty)$, the persistence diagram weakly converges to a limiting point process without normalization. Finally, if $n^{k+2}r_n^{d(k+1)} \to 0$, the sequence of probability distributions of a persistence diagram should be normalized, and the resulting convergence will be treated in terms of the $\mathcal M_0$-topology.
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Functional strong law of large numbers for Betti numbers in the tail

2021-01-01
Takashi Owada , Zifu Wei
The objective of this paper is to investigate the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers 
 The objective of this paper is to investigate the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers for Betti numbers, a basic quantifier of algebraic topology, of a geometric complex outside an open ball of radius $R_n$, such that $R_n\to\infty$ as the sample size $n$ increases. The nature of the obtained law of large numbers is determined by the decay rate of a probability density. It especially depends on whether the tail of a density decays at a regularly varying rate or an exponentially decaying rate. The nature of the limit theorem depends also on how rapidly $R_n$ diverges. In particular, if $R_n$ diverges sufficiently slowly, the limiting function in the law of large numbers is crucially affected by the emergence of arbitrarily large connected components supporting topological cycles in the limit.
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Convergence of persistence diagrams for topological crackle

2020-04-27
Takashi Owada , Omer Bobrowski
In this paper, we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the 
 In this paper, we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of topological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to topological cycles generated by random points far away from the bulk of other points, when the support is unbounded. We establish weak convergence results for persistence diagrams – a point process representation for persistent homology, where each topological cycle is represented by its $({\mathit{birth},\mathit{death}})$ coordinates. In this work, we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. Using this framework, we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a shorter lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.
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A functional non-central limit theorem for multiple-stable processes with long-range dependence

2020-04-24
Shuyang Bai , Takashi Owada , Yizao Wang
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Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex

2020-01-01
Takashi Owada , Gennady Samorodnitsky , Gugan Thoppe
Topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since the randomness there is usually 
 Topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since the randomness there is usually governed by a single parameter. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex and, more importantly, of its dynamic analogue that we introduce here. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. The dynamic versions of the clique complex and the Linial-Meshulum complex are special cases of our setup. Our key result concerns the regime where face-counts of a particular dimension dominate. We show that the Betti numbers corresponding to this dimension and the Euler characteristic satisfy functional strong law of large numbers and functional central limit theorems. Surprisingly, in the latter result, the limiting Gaussian process depends only upon the dynamics in the smallest non-trivial dimension.
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Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

2020-01-01
Andrew M. Thomas , Takashi Owada
To recover the topology of a manifold in the presence of heavy tailed or exponentially decaying noise, one must understand the behavior of geometric complexes whose points lie in the 
 To recover the topology of a manifold in the presence of heavy tailed or exponentially decaying noise, one must understand the behavior of geometric complexes whose points lie in the tail of these noise distributions. This study advances this line of inquiry, and demonstrates functional strong laws of large numbers for the Euler characteristic process of random geometric complexes formed by random points outside of an expanding ball in $\mathbb{R}^d$. When the points are drawn from a heavy tailed distribution with a regularly varying tail, the Euler characteristic process grows at a regularly varying rate, and the scaled process converges uniformly and almost surely to a smooth function. When the points are drawn from a distribution with an exponentially decaying tail, the Euler characteristic process grows logarithmically, and the scaled process converges to another smooth function in the same sense. All of the limit theorems take place when the points inside the expanding ball are densely distributed, so that the simplex counts outside of the ball of all dimensions contribute to the Euler characteristic process.

Topological crackle of heavy-tailed moving average processes

2019-01-11
Takashi Owada

A functional non-central limit theorem for multiple-stable processes with long-range dependence

2019-01-01
Shuyang Bai , Takashi Owada , Yizao Wang
A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using 
 A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets. The joint moments of the local times are computed, which play the key in the proof and are also of independent interest.
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Functional limit theorems for the Euler characteristic process in the critical regime

2019-01-01
Andrew M. Thomas , Takashi Owada
This study presents functional limit theorems for the Euler characteristic of Vietoris-Rips complexes. The points are drawn from a non-homogeneous Poisson process on $\mathbb{R}^d$, and the connectivity radius governing the 
 This study presents functional limit theorems for the Euler characteristic of Vietoris-Rips complexes. The points are drawn from a non-homogeneous Poisson process on $\mathbb{R}^d$, and the connectivity radius governing the formation of simplices is taken as a function of time parameter $t$, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have non-trivial topology. We establish two "functional-level" limit theorems, a strong law of large numbers and a central limit theorem for the appropriately normalized Euler characteristic process.
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Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

2018-08-28
Takashi Owada
We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic 
 We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic topology. It then turns out that the growth rate of the Betti numbers and the properties of the limiting processes all depend on the distance of the region of interest from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes. We also derive the limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology.
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Limit theorems for process-level Betti numbers for sparse, critical, and Poisson regimes

2018-01-01
Takashi Owada , Andrew M. Thomas
The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the $d$-dimensional Euclidean 
 The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the $d$-dimensional Euclidean space $\mathbb{R}^d$. We consider the case where the points of the Čech complex are generated by a Poisson process with intensity $nf$ for a probability density $f$. We look at the cases where the behavior of the connectivity radius of Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse and Poisson regimes, as well when the connectivity radius is on the order of $n^{-1/d}$, the critical regime. We establish limit theorems in all of the aforementioned regimes, a central limit theorem for the sparse and critical regimes, and a Poisson limit theorem for the Poisson regime. When the connectivity radius of the Čech complex is $o(n^{-1/d})$, i.e., the sparse and Poisson regimes, we can decompose the limiting processes into a time-changed Brownian motion and a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.

Convergence of Persistence Diagrams for Topological Crackle

2018-01-01
Takashi Owada , Omer Bobrowski
In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the 
 In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction of homological cycles (generalizations of loops or holes) in different dimensions. Topological crackle is a term that refers to homological cycles generated by "noisy" samples where the support is unbounded. We aim to establish weak convergence results for persistence diagrams - a point process representation for persistent homology, where each homological cycle is represented by its (birth,death) coordinates. In this work we treat persistence diagrams as random closed sets, so that the resulting weak convergence is defined in terms of the Fell topology. In this framework we show that the limiting persistence diagrams can be divided into two parts. The first part is a deterministic limit containing a densely-growing number of persistence pairs with a short lifespan. The second part is a two-dimensional Poisson process, representing persistence pairs with a longer lifespan.
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Sub-tree counts on hyperbolic random geometric graphs

2018-01-01
Takashi Owada , D. Yogeshwaran
We study the hyperbolic random geometric graph introduced in Krioukov et al. For a sequence $R_n \to \infty$, we define these graphs to have the vertex set as Poisson points 
 We study the hyperbolic random geometric graph introduced in Krioukov et al. For a sequence $R_n \to \infty$, we define these graphs to have the vertex set as Poisson points distributed uniformly in balls $B(0,R_n) \subset B_d^{\alpha}$, the $d$-dimensional Poincar\'e ball (unit d-ball with the Poincar\'e metric $d_{\alpha}$ corresponding to negative curvature $-\alpha^2, \alpha > 0$) by connecting any two points within a distance $R_n$ according to the metric $d_{\zeta}, \zeta > 0$. Denoting these graphs by $HG_n(R_n ; \alpha, \zeta)$, we study asymptotic counts of copies of a fixed tree $\Gamma_k$ (with the ordered degree sequence $d_{(1)} \leq \ldots \leq d_{(k)}$) in $HG_n(R_n ; \alpha, \zeta)$. Unlike earlier works, we count more involved structures, allowing for $d > 2$, and in many places, more general choices of $R_n$ rather than $R_n = 2[\zeta (d-1)]^{-1}\log (n/ \nu), \nu \in (0,\infty)$. The latter choice of $R_n$ for $\alpha / \zeta > 1/2$ corresponds to the thermodynamic regime. We show multiple phase transitions in $HG_n(R_n ; \alpha, \zeta)$ as $\alpha / \zeta$ increases, i.e., the space $B_d^{\alpha}$ becomes more hyperbolic. In particular, our analyses reveal that the sub-tree counts exhibit an intricate dependence on the degree sequence $d_{(1)},\ldots,d_{(k)}$ of $\Gamma_k$ as well as the ratio $\alpha/\zeta$. Under a more general radius regime $R_n$ than that described above, we investigate the asymptotics of the expectation and variance of sub-tree counts. Moreover, we prove the corresponding central limit theorem as well. Our proofs rely crucially on a careful analysis of the sub-tree counts near the boundary using Palm calculus for Poisson point processes along with estimates for the hyperbolic metric and measure. For the central limit theorem, we use the abstract normal approximation result from Last et al. derived using the Malliavin-Stein method.
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Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows

2017-07-01
Paul Jung , Takashi Owada , Gennady Samorodnitsky
We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its 
 We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its long memory which is quantified by a Mittag–Leffler process induced by an associated Harris chain, at the discrete-time level. Previous results in Owada and Samorodnitsky [Ann. Probab. 43 (2015) 240–285] dealt with positive dependence in the increment process, whereas this paper derives the functional limit theorems under negative dependence. The negative dependence is due to cancellations arising from Gaussian-type fluctuations of functionals of the associated Harris chain. The new types of limiting processes involve stable random measures, due to heavy tails, Mittag–Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations. Along the way, we prove a function central limit theorem for fluctuations of functionals of Harris chains which is of independent interest as it extends a result of Chen [Probab. Theory Related Fields 116 (2000) 89–123].
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Limit theorems for point processes under geometric constraints (and topological crackle)

2017-05-01
Takashi Owada , Robert J. Adler
We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A 
 We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Čech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via nonhomogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered. The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological "crackle;" the continued presence of spurious homology in samples of topological structures, despite increased sample size.
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Functional central limit theorem for subgraph counting processes

2017-01-01
Takashi Owada
The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The 
 The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.
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Limit Theorems for the Sum of Persistence Barcodes

2016-01-01
Takashi Owada
Topological Data Analysis (TDA) refers to an approach that uses concepts from algebraic topology to study the "shapes" of datasets. The main focus of this paper is persistent homology, a 
 Topological Data Analysis (TDA) refers to an approach that uses concepts from algebraic topology to study the "shapes" of datasets. The main focus of this paper is persistent homology, a ubiquitous tool in TDA. Basing our study on this, we investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb R^d$. In particular, we establish various limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology. It then turns out that the growth rate of the sum of the bar lengths and the properties of the limiting processes all depend on the distance of the region of interest in $\mathbb R^d$ from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes.

Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments

2015-05-27
Takashi Owada , Gennady Samorodnitsky
We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-stable process. The length of memory in the stable process is parameterized by 
 We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-stable process. The length of memory in the stable process is parameterized by a certain ergodic-theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fréchet process. It is a self-similar process with $\alpha$-Fréchet marginals, and it has stationary max-increments, a property which we introduce in this paper. The functional limit theorem is established in the space $D[0,\infty)$ equipped with the Skorohod $M_{1}$-topology; in certain special cases the topology can be strengthened to the Skorohod $J_{1}$-topology.
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Functional Central Limit Theorem for Subgraph Counting Processes

2015-01-01
Takashi Owada
The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific 
 The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.
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Limit Theorems for Point Processes under Geometric Constraints (and Topological Crackle)

2015-01-01
Takashi Owada , Robert J. Adler
We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A 
 We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of \v{C}ech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via non-homogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered. The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological `crackle'; the continued presence of spurious homology in samples of topological structures, despite increased sample size.
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Functional central limit theorem for negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows

2015-01-01
Paul Jung , Takashi Owada , Gennady Samorodnitsky
We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only 
 We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only positive dependence could be treated. The negative dependence involves cancellations of the Gaussian second order. This leads to new types of limiting processes involving stable random measures, due to heavy tails, Mittag-Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations.

Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows

2014-11-12
Takashi Owada , Gennady Samorodnitsky
We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying LĂ©vy measures. The limit process is a 
 We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying LĂ©vy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments that coincides on a part of its parameter space with a previously described process. The normalizing sequence and the limiting process are determined by the ergodic-theoretical properties of the flow underlying the integral representation of the process. These properties can be interpreted as determining how long the memory of the stationary infinitely divisible process is. We also establish functional convergence, in a strong distributional sense, for conservative pointwise dual ergodic maps preserving an infinite measure.
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Limit Theory for the Sample Autocovariance for Heavy-Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows

2014-07-02
Takashi Owada
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Limit Theory for the Sample Autocovariance for Heavy Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows

2013-01-01
Takashi Owada
This study aims to develop the limit theorems on the sample autocovariances and sample autocorrelations for certain stationary infinitely divisible processes. We consider the case where the infinitely divisible process 
 This study aims to develop the limit theorems on the sample autocovariances and sample autocorrelations for certain stationary infinitely divisible processes. We consider the case where the infinitely divisible process has heavy tail marginals and is generated by a conservative flow. Interestingly, the growth rate of the sample autocovariances is determined by not only heavy tailedness of the marginals but also memory length of the process. Although this feature was first observed by \cite{resnick:samorodnitsky:xue:2000} for some very specific processes, we will propose a more general framework from the viewpoint of infinite ergodic theory. Consequently, the asymptotics of the sample autocovariances can be more comprehensively discussed.

Empirical Likelihood Estimation of Levy Processes (Revised in March 2005)

2004-01-01
Naoto Kunitomo , Takashi Owada
We propose a new parameter estimation procedure for the Levy processes and the class of infinitely divisible distribution. We shall show that the empirical likelihood method gives an easy way 
 We propose a new parameter estimation procedure for the Levy processes and the class of infinitely divisible distribution. We shall show that the empirical likelihood method gives an easy way to estimate the key parameters of the infinitely divisible distributions including the class of stable distributions as a special case. The maximum empirical likelihood estimator by using the empirical characteristic functions gives the consistency, the asymptotic normality, and the asymptotic efficiency for the key parameters when the number of restrictions on the empirical characteristic functions is large. Test procedures can be also developed. Some extensions to the estimating equations problem with the infinitely divisible distributions are discussed.
Coauthor Papers Together
Gennady Samorodnitsky 10
Christian Hirsch 7
Andrew M. Thomas 5
Zifu Wei 4
Gugan Thoppe 2
Moritz Otto 2
D. Yogeshwaran 2
Paul Jung 2
Omer Bobrowski 2
Taegyu Kang 2
D. Yogeshwaran 2
Yizao Wang 2
Robert J. Adler 2
Shuyang Bai 2
Christoph ThÀle 2
Naoto Kunitomo 1

Limit theorems for Betti numbers of random simplicial complexes

2013-01-01
Matthew Kahle , Elizabeth Meckes
There have been several recent articles studying homology of various types of random simplicial complexes.Several theorems have concerned thresholds for vanishing of homology groups, and in some cases expectations of 
 There have been several recent articles studying homology of various types of random simplicial complexes.Several theorems have concerned thresholds for vanishing of homology groups, and in some cases expectations of the Betti numbers; however, little seems known so far about limiting distributions of random Betti numbers.In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complexes: ErdƑs-RĂ©nyi random clique complexes, random Vietoris-Rips complexes, and random Čech complexes.These results may be of practical interest in topological data analysis.
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Random geometric complexes in the thermodynamic regime

2015-11-14
D. Yogeshwaran , Eliran Subag , Robert J. Adler
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The topology of probability distributions on manifolds

2014-03-21
Omer Bobrowski , Sayan Mukherjee
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Random Geometric Complexes

2011-01-18
Matthew Kahle
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On the topology of random complexes built over stationary point processes

2015-10-01
D. Yogeshwaran , Robert J. Adler
There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of 
 There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a random point process in $\mathbb {R}^d$, and the edges and faces are determined according to some deterministic rule, typically leading to Čech and Vietoris-Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either i.i.d. observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction or repulsion, we find phenomena quantitatively different from those observed in the i.i.d. and Poisson cases. From the point of view of topological data analysis, our results seriously impact considerations of model (non)robustness for statistical inference. Our proofs rely on analysis of subgraph and component counts of stationary point processes, which are of independent interest in stochastic geometry.
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Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry

2016-05-01
Laurent Decreusefond , Matthias Schulte , Christoph ThÀle
A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on 
 A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on the target space of $f$. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived, and examples from stochastic geometry are investigated.
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Barcodes: The persistent topology of data

2007-10-26
Robert Ghrist
This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool 
 This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets—
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Convergence of Probability Measures

1999-07-16
Patrick Billingsley
Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index. Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.
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Limit theorems for persistence diagrams

2018-08-28
Yasuaki Hiraoka , Tomoyuki Shirai , Khanh Duy Trinh
The persistent homology of a stationary point process on $\mathbf{R}^{N}$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the 
 The persistent homology of a stationary point process on $\mathbf{R}^{N}$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.
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Limit theorems for point processes under geometric constraints (and topological crackle)

2017-05-01
Takashi Owada , Robert J. Adler
We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A 
 We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Čech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via nonhomogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered. The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological "crackle;" the continued presence of spurious homology in samples of topological structures, despite increased sample size.
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Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes

2018-08-28
Takashi Owada
We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic 
 We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic topology. It then turns out that the growth rate of the Betti numbers and the properties of the limiting processes all depend on the distance of the region of interest from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes. We also derive the limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology.
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The scaling limit of Poisson-driven order statistics with applications in geometric probability

2012-09-03
Matthias Schulte , Christoph ThÀle
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Distance functions, critical points, and the topology of random Čech complexes

2014-01-01
Omer Bobrowski , Robert J. Adler
For a finite set of points P in R d , the function d P : R d → R + measures Euclidean distance to the set P. We study 
 For a finite set of points P in R d , the function d P : R d → R + measures Euclidean distance to the set P. We study the number of critical points of d P when P is a Poisson process.In particular, we study the limit behavior of N k -the number of critical points of d P with Morse index k-as the density of points grows.We present explicit computations for the normalized limiting expectations and variances of the N k , as well as distributional limit theorems.We link these results to recent results in [16,17] in which the Betti numbers of the random Čech complex based on P were studied.
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Topology of random geometric complexes: a survey

2018-01-23
Omer Bobrowski , Matthew Kahle
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Functional central limit theorem for subgraph counting processes

2017-01-01
Takashi Owada
The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The 
 The objective of this study is to investigate the limiting behavior of a subgraph counting process built over random points from an inhomogeneous Poisson point process on $\mathbb R^d$. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.
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Homology of Multi-Parameter Random Simplicial Complexes

2019-04-26
Christopher F. Fowler

Random Measures, Theory and Applications

2017-01-01
Olav Kallenberg

Randomly Weighted $d$-Complexes: Minimal Spanning Acycles and Persistence Diagrams

2020-04-17
PrimoĆŸ Ć kraba , Gugan Thoppe , D. Yogeshwaran
A weighted $d$-complex is a simplicial complex of dimension $d$ in which each face is assigned a real-valued weight. We derive three key results here concerning persistence diagrams and minimal 
 A weighted $d$-complex is a simplicial complex of dimension $d$ in which each face is assigned a real-valued weight. We derive three key results here concerning persistence diagrams and minimal spanning acycles (MSAs) of such complexes. First, we establish an equivalence between the MSA face-weights and death times in the persistence diagram. Next, we show a novel stability result for the MSA face-weights which, due to our first result, also holds true for the death and birth times, separately. Our final result concerns a perturbation of a mean-field model of randomly weighted $d$-complexes. The $d$-face weights here are perturbations of some i.i.d. distribution while all the lower-dimensional faces have a weight of $0$. If the perturbations decay sufficiently quickly, we show that suitably scaled extremal nearest face-weights, face-weights of the $d$-MSA, and the associated death times converge to an inhomogeneous Poisson point process. This result completely characterizes the extremal points of persistence diagrams and MSAs. The point process convergence and the asymptotic equivalence of three point processes are new for any weighted random complex model, including even the non-perturbed case. Lastly, as a consequence of our stability result, we show that Frieze's $\zeta(3)$ limit for random minimal spanning trees and the recent extension to random MSAs by Hino and Kanazawa also hold in suitable noisy settings.
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Poisson limits for U-statistics

2002-05-01
André Dabrowski , Herold Dehling , Thomas Mikosch +1 more

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

2018-12-10
Akshay Goel , Khanh Duy Trinh , Kenkichi Tsunoda
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The Shape of Asymptotic Dependence

2012-11-10
Guus Balkema , Paul Embrechts , Natalia Nolde

A Topological View of Unsupervised Learning from Noisy Data

2011-01-01
Partha Niyogi , Stephen T. Smale , Shmuel Weinberger
In this paper, we take a topological view of unsupervised learning. From this point of view, clustering may be interpreted as trying to find the number of connected components of 
 In this paper, we take a topological view of unsupervised learning. From this point of view, clustering may be interpreted as trying to find the number of connected components of any underlying geometrically structured probability distribution in a certain sense that we will make precise. We construct a geometrically structured probability distribution that seems appropriate for modeling data in very high dimensions. A special case of our construction is the mixture of Gaussians where there is Gaussian noise concentrated around a finite set of points (the means). More generally we consider Gaussian noise concentrated around a low dimensional manifold and discuss how to recover the homology of this underlying geometric core from data that do not lie on it. We show that if the variance of the Gaussian noise is small in a certain sense, then the homology can be learned with high confidence by an algorithm that has a weak (linear) dependence on the ambient dimension. Our algorithm has a natural interpretation as a spectral learning algorithm using a combinatorial Laplacian of a suitable data-derived simplicial complex.
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Computing Persistent Homology

2004-11-19
Afra Zomorodian , Gunnar Carlsson
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Coverage in sensor networks via persistent homology

2007-04-25
Vin de Silva , Robert Ghrist
We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks 
 We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks with minimal sensing capabilities).In particular, we give a homological criterion to rigorously guarantee that a collection of balls covers a bounded domain based on the homology of a certain simplicial pair.This pair of (Vietoris-Rips) complexes is derived from graphs representing a coarse form of distance estimation between nodes and a proximity sensor for the boundary of the domain.The methods we introduce come from persistent homology theory and are applicable to nonlocalized sensor networks with ad hoc wireless communications.
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The asymptotic distributional behaviour of transformations preserving infinite measures

1981-12-01
Jon Aaronson

Sharp vanishing thresholds for cohomology of random flag complexes

2014-02-20
Matthew Kahle
For every k ≄ 1, the k-th cohomology group H k (X, Q) of the random flag complex X ∌ X(n, p) passes through two phase transitions: one where it 
 For every k ≄ 1, the k-th cohomology group H k (X, Q) of the random flag complex X ∌ X(n, p) passes through two phase transitions: one where it appears and one where it vanishes.We describe the vanishing threshold and show that it is sharp.Using the same spectral methods, we also find a sharp threshold for the fundamental group π1(X) to have Kazhdan's property (T).Combining with earlier results, we obtain as a corollary that for every k ≄ 3, there is a regime in which the random flag complex is rationally homotopy equivalent to a bouquet of k-dimensional spheres.
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On the Structure of Stationary Stable Processes

1995-07-01
J. RosiƄski
A connection between structural studies of stationary non-Gaussian stable processes and the ergodic theory of nonsingular flows is established and exploited. Using this connection, a unique decomposition of a stationary 
 A connection between structural studies of stationary non-Gaussian stable processes and the ergodic theory of nonsingular flows is established and exploited. Using this connection, a unique decomposition of a stationary stable process into three independent stationary parts is obtained. It is shown that the dissipative part of a flow generates a mixed moving average part of a stationary stable process, while the identity part of a flow essentially gives the harmonizable part. The third part of a stationary process is determined by a conservative flow without fixed points and by a related cocycle.
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Topological crackle of heavy-tailed moving average processes

2019-01-11
Takashi Owada

Null flows, positive flows and the structure of stationary symmetric stable processes

2005-09-01
Gennady Samorodnitsky
This paper elucidates the connection between stationary symmetric α-stable processes with 0<α<2 and nonsingular flows on measure spaces by describing a new and unique decomposition of stationary stable processes into 
 This paper elucidates the connection between stationary symmetric α-stable processes with 0<α<2 and nonsingular flows on measure spaces by describing a new and unique decomposition of stationary stable processes into those corresponding to positive flows and those corresponding to null flows. We show that necessary and sufficient for a stationary stable process to be ergodic is that its positive component vanishes.
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Polar decomposition of regularly varying time series in star-shaped metric spaces

2017-02-24
Johan Segers , Yuwei Zhao , Thomas Meinguet
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Persistent homology for random fields and complexes

2010-01-01
Robert J. Adler , Omer Bobrowski , Matthew Strom Borman +2 more
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more 
 We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the algebraic topology of the excursion sets of random fields.
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Growth rates of sample covariances of stationary symmetric α-stable processes associated with null recurrent Markov chains

2000-02-01
Sidney I. Resnick , Gennady Samorodnitsky , Fang Xue
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Regular variation for measures on metric spaces

2006-01-01
Henrik Hult , Filip Lindskog
The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate 
 The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regular variation is defined in this general setting and several statements that are equivalent to this definition are presented. This extends the notion of regular variation for Borel measures on the Euclidean space Rd to more general metric spaces. Some examples, including regular variation for Borel measures on Rd, the space of continuous functions C and the Skorohod space D, are provided.
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Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

2017-11-10
Sascha Bachmann , Matthias Reitzner
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Foundations of Modern Probability

2021-01-01
Olav Kallenberg
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Convergence of Probability Measures

1970-02-01
James L. Snell
Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index. Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.

Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps

2014-01-01
Filip Lindskog , Sidney I. Resnick , Joyjit Roy
We develop a framework for regularly varying measures on complete separable metric spaces $\mathbb{S}$ with a closed cone $\mathbb{C}$ removed, extending material in [15,24]. Our framework provides a flexible way 
 We develop a framework for regularly varying measures on complete separable metric spaces $\mathbb{S}$ with a closed cone $\mathbb{C}$ removed, extending material in [15,24]. Our framework provides a flexible way to consider hidden regular variation and allows simultaneous regular-variation properties to exist at different scales and provides potential for more accurate estimation of probabilities of risk regions. We apply our framework to iid random variables in $\mathbb{R}_{+}^{\infty}$ with marginal distributions having regularly varying tails and to cĂ dlĂ g LĂ©vy processes whose LĂ©vy measures have regularly varying tails. In both cases, an infinite number of regular-variation properties coexist distinguished by different scaling functions and state spaces.
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Limit Theory for Moving Averages of Random Variables with Regularly Varying Tail Probabilities

1985-02-01
Richard A. Davis , Sidney I. Resnick
Let $\{Z_k, -\infty < k < \infty\}$ be iid where the $Z_k$'s have regularly varying tail probabilities. Under mild conditions on a real sequence $\{c_j, j \geq 0\}$ the stationary 
 Let $\{Z_k, -\infty < k < \infty\}$ be iid where the $Z_k$'s have regularly varying tail probabilities. Under mild conditions on a real sequence $\{c_j, j \geq 0\}$ the stationary process $\{X_n: = \sum^\infty_{j=0} c_jZ_{n-j}, n \geq 1\}$ exists. A point process based on $\{X_n\}$ converges weakly and from this, a host of weak limit results for functionals of $\{X_n\}$ ensue. We study sums, extremes, excedences and first passages as well as behavior of sample covariance functions.
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Large Deviations for Point Processes Based on Stationary Sequences with Heavy Tails

2010-03-01
Henrik Hult , Gennady Samorodnitsky
In this paper we propose a framework that facilitates the study of large deviations for point processes based on stationary sequences with regularly varying tails. This framework allows us to 
 In this paper we propose a framework that facilitates the study of large deviations for point processes based on stationary sequences with regularly varying tails. This framework allows us to keep track both of the magnitude of the extreme values of a process and the order in which these extreme values appear. Particular emphasis is put on (infinite) linear processes with random coefficients. The proposed framework provides a fairly complete description of the joint asymptotic behavior of the large values of the stationary sequence. We apply the general result on large deviations for point processes to derive the asymptotic decay of certain probabilities related to partial sum processes as well as ruin probabilities.
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An Introduction to Infinite Ergodic Theory

1997-05-13
Jon Aaronson

Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments

2015-05-27
Takashi Owada , Gennady Samorodnitsky
We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-stable process. The length of memory in the stable process is parameterized by 
 We derive a functional limit theorem for the partial maxima process based on a long memory stationary $\alpha$-stable process. The length of memory in the stable process is parameterized by a certain ergodic-theoretical parameter in an integral representation of the process. The limiting process is no longer a classical extremal Fréchet process. It is a self-similar process with $\alpha$-Fréchet marginals, and it has stationary max-increments, a property which we introduce in this paper. The functional limit theorem is established in the space $D[0,\infty)$ equipped with the Skorohod $M_{1}$-topology; in certain special cases the topology can be strengthened to the Skorohod $J_{1}$-topology.
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Second-order properties and central limit theorems for geometric functionals of Boolean models

2016-01-05
Daniel Hug , GĂŒnter Last , Matthias Schulte
Let $Z$ be a Boolean model based on a stationary Poisson process $\eta$ of compact, convex particles in Euclidean space ${\mathbb{R}}^d$. Let $W$ denote a compact, convex observation window. For 
 Let $Z$ be a Boolean model based on a stationary Poisson process $\eta$ of compact, convex particles in Euclidean space ${\mathbb{R}}^d$. Let $W$ denote a compact, convex observation window. For a large class of functionals $\psi$, formulas for mean values of $\psi(Z\cap W)$ are available in the literature. The first aim of the present work is to study the asymptotic covariances of general geometric (additive, translation invariant and locally bounded) functionals of $Z\cap W$ for increasing observation window $W$, including convergence rates. Our approach is based on the Fock space representation associated with $\eta$. For the important special case of intrinsic volumes, the asymptotic covariance matrix is shown to be positive definite and can be explicitly expressed in terms of suitable moments of (local) curvature measures in the isotropic case. The second aim of the paper is to prove multivariate central limit theorems including Berry-Esseen bounds. These are based on a general normal approximation result obtained by the Malliavin--Stein method.
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Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows

2014-11-12
Takashi Owada , Gennady Samorodnitsky
We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying LĂ©vy measures. The limit process is a 
 We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying LĂ©vy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments that coincides on a part of its parameter space with a previously described process. The normalizing sequence and the limiting process are determined by the ergodic-theoretical properties of the flow underlying the integral representation of the process. These properties can be interpreted as determining how long the memory of the stationary infinitely divisible process is. We also establish functional convergence, in a strong distributional sense, for conservative pointwise dual ergodic maps preserving an infinite measure.
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Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points

2000-10-01
Roland ZweimĂŒller
We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do 
 We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do not assume the existence of a Markov partition but only require the first image of the fundamental partition to be finite, we use canonical Markov extensions to first prove pointwise dual-ergodicity, which, together with an identification of wandering rates, leads to distributional limit theorems. We show that $T$ satisfies Rohlin's formula and prove a variant of the Shannon–McMillan–Breiman theorem. Moreover, we give a stronger limit theorem for the transfer operator providing us with a large collection of uniform and Darling–Kac sets. This enables us to apply recent results from fluctuation theory.

Large random simplicial complexes, III the critical dimension

2016-10-14
A. Costa , Michael FĂ€rber
In this paper, we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in [Random simplicial complexes, preprint (2014), arXiv:1412.5805 ; Large random 
 In this paper, we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in [Random simplicial complexes, preprint (2014), arXiv:1412.5805 ; Large random simplicial complexes, I, preprint (2015), arXiv:1503.06285 ; Large random simplical complexes, II, preprint (2015), arXiv:1509.04837 ]. This model includes as special cases the Linial–Meshulam–Wallach model [Homological connectivity of random 2-complexes, Combinatorica 26 (2006) 475–487; Homological connectivity of random [Formula: see text]-complexes, Random Struct. Alogrithms 34 (2009) 408–417.] as well as the clique complexes of random graphs. We characterize the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.
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On the evolution of topology in dynamic clique complexes

2016-12-01
Gugan Thoppe , D. Yogeshwaran , Robert J. Adler
Abstract We consider a time varying analogue of the ErdƑs–RĂ©nyi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are 
 Abstract We consider a time varying analogue of the ErdƑs–RĂ©nyi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve independently as continuous-time Markov chains. Our main result is that when the edge inclusion probability is of the form p = n α , where n is the number of vertices and α∈(-1/ k , -1/( k + 1)), then the process of the normalised k th Betti number of these dynamic clique complexes converges weakly to the Ornstein–Uhlenbeck process as n →∞.
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Topology of random clique complexes

2008-04-17
Matthew Kahle
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Maximally persistent cycles in random geometric complexes

2017-08-01
Omer Bobrowski , Matthew Kahle , PrimoĆŸ Ć kraba
We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree-$k$ in persistent homology, for a either the Čech 
 We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree-$k$ in persistent homology, for a either the Čech or the Vietoris–Rips filtration built on a uniform Poisson process of intensity $n$ in the unit cube $[0,1]^{d}$. This is a natural way of measuring the largest “$k$-dimensional hole” in a random point set. This problem is in the intersection of geometric probability and algebraic topology, and is naturally motivated by a probabilistic view of topological inference. We show that for all $d\ge2$ and $1\le k\le d-1$ the maximally persistent cycle has (multiplicative) persistence of order \[\Theta ((\frac{\log n}{\log\log n})^{1/k}),\] with high probability, characterizing its rate of growth as $n\to\infty$. The implied constants depend on $k$, $d$ and on whether we consider the Vietoris–Rips or Čech filtration.
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An Asymptotic Evaluation of the Tail of a Multiple Symmetric $\alpha$- Stable Integral

1989-10-01
Gennady Samorodnitsky , Jerzy Szulga
We expand a multiple symmetric $\alpha$-stable integral $\int \cdots \int f(t_1, \ldots, t_n) dM(t_1) \ldots dM(t_n)$ into a LePage type multiple series of transformed arrival times of a Poisson process. 
 We expand a multiple symmetric $\alpha$-stable integral $\int \cdots \int f(t_1, \ldots, t_n) dM(t_1) \ldots dM(t_n)$ into a LePage type multiple series of transformed arrival times of a Poisson process. An exact evaluation of the limit of appropriately normalized tail distribution results from this representation.
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Inside the critical window for cohomology of random <i>k</i> -complexes

2015-01-29
Matthew Kahle , Boris Pittel
We prove sharper versions of theorems of Linial–Meshulam and Meshulam–Wallach which describe the behavior for -cohomology of a random k-dimensional simplicial complex within a narrow transition window. In particular, we 
 We prove sharper versions of theorems of Linial–Meshulam and Meshulam–Wallach which describe the behavior for -cohomology of a random k-dimensional simplicial complex within a narrow transition window. In particular, we show that if Y is a random k-dimensional simplicial complex with each k-simplex appearing i.i.d. with probability with and fixed, then the dimension of cohomology is asymptotically Poisson distributed with mean . In the k = 2 case we also prove that in an accompanying growth process, with high probability, vanishes exactly at the moment when the last -simplex gets covered by a k-simplex, a higher-dimensional analogue of a "stopping time" theorem about connectivity of random graphs due to BollobĂĄs and Thomason. Random Struct. Alg., 2015 © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 102–124, 2016
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