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Let {Yn} be a sequence of dependent random variables and {On ( , • ) } be a sequence of Borel functions.Let On be a solution of the equation Mn …
Let {Yn} be a sequence of dependent random variables and {On ( , • ) } be a sequence of Borel functions.Let On be a solution of the equation Mn (x) = 0 for each n 1, where Mn (x) DOT/ (x, Y.).A Robbins-Monro type stochastic approximation procedure Xn+1 = X,, -anOn (Xn, Yn) is considered for estimating On for n sufficiently large.Under some assumptions about {an} {On}, {Y.} and {On (• , • )} which may not include the fundamental condition ELOn (Xn, Y.) I Xi, , Xn,]=-Mn(Xn) a.s., the a.s.convergence and in mean-square con vergence of Xn-BnI to zero are studied.
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In this paper, we will associated with our works to state some new advance in the theory of strong approximations. There are five parts in the paper, i. e. 1. …
In this paper, we will associated with our works to state some new advance in the theory of strong approximations. There are five parts in the paper, i. e. 1. Increments of a Wiener process. 2. Strong approximations for partial sums of IIDRV's. 3. The increments of partial sums of independent R. V. s (non-identically distri- buted). 4. Strong approximations of mixing R V. s. 5. Strong approximations for other kinds of dependent R. V.'s and functional of R. V. s.
Let 81, be the o-field of Borel subsets of the Euclidean space R& , f,x be the set of all probability measures on Bp, Dp be the set of infinitely …
Let 81, be the o-field of Borel subsets of the Euclidean space R& , f,x be the set of all probability measures on Bp, Dp be the set of infinitely divisible distributions in fp.The notation r € Re will indicate that o : (at,...,te) with rr'€ Rt, j:lr...rlr.For o,y € Re wewrite o <y if rr'<Vi forall j : Lr...rk.For the e-neighbourhood of a set X C R& we use the notation X': {y e Rt: inf,6lsllt-yll < e}.Wealsodenote: 1: (1,1,...,1) € Re, E E f3 the probability distribution concentrated at zero, O(.F') € f,x the Gaussian distribution having the same mean and the same covaria,nce operator as a given F € F*, "(F): e-l D;= o F* f m! (products and p-owers of measures are understoodintheconvolutionsense), F(a): F{{u e Rk: u < t}}, o € Re the comesponding distribution function.The symbols C1,C2,... will be used to denote positive constants depending only on the dimension /c.We shall estimate the following characteristics of proximity of distributions F,G e f*: the uniform distance p(F,c): jåå, lF(r) -G(r)l ; the multidimensional analogue of the Ldvy distance L(F,G): inf{e : F(r -eL)-e <G(r) < F(**e1) *e for all r e R&}; the Ldvy-Prokhorov distance r(F,G): inf{e: r{x} < G{x"} *e, G{x} < F{x"} *e for all x e Bs} and for .\> 0 sup max{ F(" + 0) -G(r * s€Re sup maxtr{x} -G{x^}, X e6nNow we pass on to the statement of the problem.We consider the convolution
The joint statistics of partial sums of ordered random variables (RVs) are often needed for the accurate performance characterization of a wide variety of wireless communication systems. A unified analytical …
The joint statistics of partial sums of ordered random variables (RVs) are often needed for the accurate performance characterization of a wide variety of wireless communication systems. A unified analytical framework to determine the joint statistics of partial sums of ordered independent and identically distributed (i.i.d.) random variables was recently presented. However, the identical distribution assumption may not be valid in several real-world applications. With this motivation in mind, we consider in this paper the more general case in which the random variables are independent but not necessarily identically distributed (i.n.d.). More specifically, we extend the previous analysis and introduce a new more general unified analytical framework to determine the joint statistics of partial sums of ordered i.n.d. RVs. Our mathematical formalism is illustrated with an application on the exact performance analysis of the capture probability of generalized selection combining (GSC)-based RAKE receivers operating over frequency-selective fading channels with a non-uniform power delay profile.
In this paper the computational aspects of probability calculations for dynamical partial sum expressions are discussed. Such dynamical partial sum expressions have many important applications, and examples are provided in …
In this paper the computational aspects of probability calculations for dynamical partial sum expressions are discussed. Such dynamical partial sum expressions have many important applications, and examples are provided in the fields of reliability, product quality assessment, and stochastic control. While these probability calculations are ostensibly of a high dimension, and consequently intractable in general, it is shown how a recursive integration methodology can be implemented to obtain exact calculations as a series of two-dimensional calculations. The computational aspects of the implementaion of this methodology, with the adoption of Fast Fourier Transforms, are discussed.
Abstract We introduce a new method for high-dimensional, online changepoint detection in settings where a p-variate Gaussian data stream may undergo a change in mean. The procedure works by performing …
Abstract We introduce a new method for high-dimensional, online changepoint detection in settings where a p-variate Gaussian data stream may undergo a change in mean. The procedure works by performing likelihood ratio tests against simple alternatives of different scales in each coordinate, and then aggregating test statistics across scales and coordinates. The algorithm is online in the sense that both its storage requirements and worst-case computational complexity per new observation are independent of the number of previous observations; in practice, it may even be significantly faster than this. We prove that the patience, or average run length under the null, of our procedure is at least at the desired nominal level, and provide guarantees on its response delay under the alternative that depend on the sparsity of the vector of mean change. Simulations confirm the practical effectiveness of our proposal, which is implemented in the R package ocd, and we also demonstrate its utility on a seismology data set.
Gaussian couplings of partial sum processes are derived for the high-dimensional regime d=o(n1∕3). The coupling is derived for sums of independent random vectors and subsequently extended to nonstationary time series. …
Gaussian couplings of partial sum processes are derived for the high-dimensional regime d=o(n1∕3). The coupling is derived for sums of independent random vectors and subsequently extended to nonstationary time series. Our inequalities depend explicitly on the dimension and on a measure of nonstationarity, and are thus also applicable to arrays of random vectors. To enable high-dimensional statistical inference, a feasible Gaussian approximation scheme is proposed. Applications to sequential testing and change-point detection are described.
Survival analysis has become in a common procedure in biomedical re- searches. Conventionally, the well-known nonparametric Kaplan-Meier (KM) estimator is used in order to approximate the real survivor curve. However, …
Survival analysis has become in a common procedure in biomedical re- searches. Conventionally, the well-known nonparametric Kaplan-Meier (KM) estimator is used in order to approximate the real survivor curve. However, in competing risk contexts where more than one failure cause compete to occur and only one of them is of interest, the direct use of the Kaplan-Meier statistic does not perform correctly and, in or- der to obtain a good estimation, it must be adapted. In this work, via Monte Carlo simulations, the author explores the behavior of the Kaplan-Meier estimator in a competing risk context. In addition, dif- ferences between KM and multiple decrement methods are pointed out. Finally, a real-data problem is used in order to illustrate the situation.
A copolymer is a chain of repetitive units (monomers) that are almost identical, but they differ in their degree of affinity for certain solvents. This difference leads to striking phenomena …
A copolymer is a chain of repetitive units (monomers) that are almost identical, but they differ in their degree of affinity for certain solvents. This difference leads to striking phenomena when the polymer fluctuates in a nonhomogeneous medium, for example, made of two solvents separated by an interface. One may observe, for instance, the localization of the polymer at the interface between the two solvents. A discrete model of such system, based on the simple symmetric random walk on ℤ, has been investigated in [8], notably in the weak polymer-solvent coupling limit, where the convergence of the discrete model toward a continuum model, based on Brownian motion, has been established. This result is remarkable because it strongly suggests a universal feature of copolymer models. In this work, we prove that this is indeed the case. More precisely, we determine the weak coupling limit for a general class of discrete copolymer models, obtaining as limits a one-parameter [α ∈ (0, 1)] family of continuum models, based on α-stable regenerative sets.
In this paper, we provide the strong approximation of normalized empirical copula processes by a Gaussian process. In addition, we establish a strong approximation of the smoothed empirical copula processes …
In this paper, we provide the strong approximation of normalized empirical copula processes by a Gaussian process. In addition, we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm.
Consider the directed process $(i, S_i)$ where the second component is simple random walk on $\mathbb{Z} (S_0 = 0)$. Define a transformed path measure by weighting each $n$-step path with …
Consider the directed process $(i, S_i)$ where the second component is simple random walk on $\mathbb{Z} (S_0 = 0)$. Define a transformed path measure by weighting each $n$-step path with a factor $\exp [\lambda \sum_{1 \leq i \leq n}(\omega_i + h)\sign (S_i)]$. Here, $(\omega_i)_{i \geq 1}$ is an i.i.d. sequence of random variables taking values $\pm 1$ with probability 1/2 (acting as a random medium) , while $\lambda \in [0, \infty)$ and $h \in [0, 1)$ are parameters. The weight factor has a tendency to pull the path towards the horizontal, because it favors the combinations $S_i > 0, \omega_i = +1$ and $S_i < 0, \omega_i = -1$. The transformed path measure describes a heteropolymer, consisting of hydrophylic and hydrophobic monomers, near an oil-water interface. We study the free energy of this model as $n \to \infty$ and show that there is a critical curve $\lambda \to h_c (\lambda)$ where a phase transition occurs between localized and delocalized behavior (in the vertical direction). We derive several properties of this curve, in particular, its behavior for $\lambda \downarrow 0$. To obtain this behavior, we prove that as $\lambda, h \downarrow 0$ the free energy scales to its Brownian motion analogue.
The celebrated results of Koml\'os, Major and Tusn\'ady [Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58] give optimal Wiener approximation for the partial sums …
The celebrated results of Koml\'os, Major and Tusn\'ady [Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58] give optimal Wiener approximation for the partial sums of i.i.d. random variables and provide a powerful tool in probability and statistics. In this paper we extend KMT approximation for a large class of dependent stationary processes, solving a long standing open problem in probability theory. Under the framework of stationary causal processes and functional dependence measures of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150-14154], we show that, under natural moment conditions, the partial sum processes can be approximated by Wiener process with an optimal rate. Our dependence conditions are mild and easily verifiable. The results are applied to ergodic sums, as well as to nonlinear time series and Volterra processes, an important class of nonlinear processes.
We propose a general purpose confidence interval procedure (CIP) for statistical functionals constructed using data from a stationary time series. The procedures we propose are based on derived distribution-free analogues …
We propose a general purpose confidence interval procedure (CIP) for statistical functionals constructed using data from a stationary time series. The procedures we propose are based on derived distribution-free analogues of the χ 2 and Student’s t random variables for the statistical functional context and hence apply in a wide variety of settings including quantile estimation, gradient estimation, M-estimation, Conditional Value at Risk (CVaR) estimation, and arrival process rate estimation, apart from more traditional statistical settings. Like the method of subsampling, we use overlapping batches (OB) of time-series data to estimate the underlying variance parameter; unlike subsampling and the bootstrap, however, we assume that the implied point estimator of the statistical functional obeys a central limit theorem (CLT) to help identify the weak asymptotics (called OB-x limits, x = I, II, III) of batched Studentized statistics. The OB-x limits, certain functionals of the Wiener process parameterized by the size of the batches and the extent of their overlap, form the essential machinery for characterizing dependence and, consequently, the correctness of the proposed CIPs. The message from extensive numerical experimentation is that in settings where a functional CLT on the point estimator is in effect, using large overlapping batches alongside OB-x critical values yields confidence intervals that are often of significantly higher quality than those obtained from more generic methods like subsampling or the bootstrap. We illustrate using examples from CVaR estimation, ARMA parameter estimation, and non-homogeneous Poisson process rate estimation; R and MATLAB code for OB-x critical values is available at web.ics.purdue.edu/∼pasupath .
Let $S_t$ be a version of the slope at time $t$ of the concave majorant of Brownian motion on $\lbrack 0, \infty)$. It is shown that the process $S = …
Let $S_t$ be a version of the slope at time $t$ of the concave majorant of Brownian motion on $\lbrack 0, \infty)$. It is shown that the process $S = \{1/S_t: t > 0\}$ is the inverse of a pure jump process with independent nonstationary increments and that Brownian motion can be generated by the latter process and Brownian excursions between values of the process at successive jump times. As an application the limiting distribution of the $L_2$-norm of the slope of the concave majorant of the empirical process is derived.
We study the almost sure asymptotic properties of the local time of the uniform empirical process. In particular, we obtain two versions of the law of the iterated logarithm for …
We study the almost sure asymptotic properties of the local time of the uniform empirical process. In particular, we obtain two versions of the law of the iterated logarithm for the integral of the square of the local time. It is interesting to note that the corresponding problems for the Wiener process remain open. Properties of Lp-norms of the local time are studied. We also characterize the joint asymptotics of the local time at a fixed level and the maximum local time.
The Erdős-Turán law gives a normal approximation for the order of a randomly chosen permutation of n objects. In this paper, we provide a sharp error estimate for the approximation, …
The Erdős-Turán law gives a normal approximation for the order of a randomly chosen permutation of n objects. In this paper, we provide a sharp error estimate for the approximation, showing that, if the mean of the approximating normal distribution is slightly adjusted, the error is of order log −1/2 n .
In this paper we study the rate of convergence of the iterates of \iid random piecewise constant monotone maps to the time-$1$ transport map for the process of coalescing Brownian …
In this paper we study the rate of convergence of the iterates of \iid random piecewise constant monotone maps to the time-$1$ transport map for the process of coalescing Brownian motions. We prove that the rate of convergence is given by a power law. The time-1 map for the coalescing Brownian motions can be viewed as a fixed point for a natural renormalization transformation acting in the space of probability laws for random piecewise constant monotone maps. Our result implies that this fixed point is exponentially stable.
In this paper, we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate $n^{1/3}$. The Grenander estimator, the nonparametric …
In this paper, we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate $n^{1/3}$. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function $f$ on $[0,\infty)$, is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for $f(t_0)$, where $t_0\in(0,\infty)$ is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical distribution function $\mathbb{F}_n$ or its least concave majorant $\tilde{F}_n$, does not have any weak limit in probability. We provide a set of sufficient conditions for the consistency of any bootstrap method in this example and show that bootstrapping from a smoothed version of $\tilde{F}_n$ leads to strongly consistent estimators. The $m$ out of $n$ bootstrap method is also shown to be consistent while generating samples from $\mathbb{F}_n$ and $\tilde{F}_n$.
We introduce a general technique for proving estimates for certain random planar maps which belong to the $\gamma$-Liouville quantum gravity (LQG) universality class for $\gamma \in (0,2)$. The family of …
We introduce a general technique for proving estimates for certain random planar maps which belong to the $\gamma$-Liouville quantum gravity (LQG) universality class for $\gamma \in (0,2)$. The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d.\ increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; $\gamma=\sqrt{8/3}$); and planar maps weighted by the number of different spanning trees ($\gamma=\sqrt 2$), bipolar orientations ($\gamma=\sqrt{4/3}$), or Schnyder woods ($\gamma=1$) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (1993) prediction for the Hausdorff dimension of $\gamma$-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map $M$ to a mated-CRT map---a random planar map constructed from a correlated two-dimensional Brownian motion---using a strong coupling (Zaitsev, 1998) of the encoding walk for $M$ and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in $M$ from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when $\gamma=\sqrt{8/3}$, we instead deduce estimates for the $\sqrt{8/3}$-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.
We prove strong approximations for partial sums indexed by a renewal process. The obtained results are optimal. The established probability inequalities are also used to get bounds for the rate …
We prove strong approximations for partial sums indexed by a renewal process. The obtained results are optimal. The established probability inequalities are also used to get bounds for the rate of convergence of some limit theorems in queueing theory.
Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. The Monte Carlo error in estimation has an asymptotic normal distribution whose multivariate nature …
Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. The Monte Carlo error in estimation has an asymptotic normal distribution whose multivariate nature has so far been ignored in the MCMC community. We present a class of multivariate spectral variance estimators for the asymptotic covariance matrix in the Markov chain central limit theorem and provide conditions for strong consistency. We examine the finite sample properties of the multivariate spectral variance estimators and its eigenvalues in the context of a vector autoregressive process of order 1.
A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar …
A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $\gamma $-Liouville quantum gravity (LQG) for $\gamma \in (0,2)$ if we take the correlation to be $-\cos (\pi \gamma ^{2}/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the quantitative properties of random walk and discrete conformal embeddings for these maps. For example, our results give an independent proof that the simple random walk on the mated-CRT map is recurrent, and a polynomial upper bound for the maximum length of the edges of the mated-CRT map under a version of the Tutte embedding. Our results are also used in other work by the first two authors which shows that for a class of random planar maps — including mated-CRT maps and the UIPT — the spectral dimension is two (i.e., the return probability of the simple random walk to its starting point after $n$ steps is $n^{-1+o_{n}(1)}$) and the typical exit time of the walk from a graph-distance ball is bounded below by the volume of the ball, up to a polylogarithmic factor.
In this paper we develop a nonparametric approach to clustering very high-dimensional data, designed particularly for problems where the mixture nature of a population is expressed through multimodality of its …
In this paper we develop a nonparametric approach to clustering very high-dimensional data, designed particularly for problems where the mixture nature of a population is expressed through multimodality of its density. Therefore, a technique based implicitly on mode testing can be particularly effective. In principle, several alternative approaches could be used to assess the extent of multimodality, but in the present problem the excess mass method has important advantages. We show that the resulting methodology for determining clusters is particularly effective in cases where the data are relatively heavy tailed or show a moderate to high degree of correlation, or when the number of important components is relatively small. Conversely, in the case of light-tailed, almost-independent components when there are many clusters, clustering in terms of modality can be less reliable than more conventional approaches. This article has supplementary material online.
We prove a uniform functional law of the logarithm for the local empirical process. To accomplish this we combine techniques from classical and abstract empirical process theory, Gaussian distributional approximation …
We prove a uniform functional law of the logarithm for the local empirical process. To accomplish this we combine techniques from classical and abstract empirical process theory, Gaussian distributional approximation and probability on Banach spaces. The body of techniques we develop should prove useful to the study of the strong consistency of d-variate kernel-type nonparametric function estimators.
Let $\{x_1, x_2,\cdots\}$ be a sequence of i.i.d.r.v. with mean zero, variance one, and (1) $\mathbf{P}(|x_k| \geqq \lambda) \leqq C \exp(-\alpha\lambda^\varepsilon)$ for positive $\alpha, \varepsilon$. Let $f(t, x)$ (with its …
Let $\{x_1, x_2,\cdots\}$ be a sequence of i.i.d.r.v. with mean zero, variance one, and (1) $\mathbf{P}(|x_k| \geqq \lambda) \leqq C \exp(-\alpha\lambda^\varepsilon)$ for positive $\alpha, \varepsilon$. Let $f(t, x)$ (with its first partial derivatives) be of slow growth in $x$, let $F_n(x)$ be the distribution function of $(1/n) \sum^n_1 f(k/n, s_k/n^{\frac{1}{2}})$ where $s_k = x_1 + x_2 + \cdots + x_k$, and let $F(x)$ be the distribution function of $\int^1_0 f(t, w(t)) dt$ where $\{w(t)\}$ is Brownian motion. Then $\sup_x |F_n(x) - F(x)| = O((\log n)^\beta/n^{\frac{1}{2}})$ provided $F(x)$ has a bounded derivative. The proof uses the Skorokhod representation; also, a theorem is proven which would indicate that the Skorokhod representation cannot be used in general to obtain a rate of convergence better than $O(1/n^{\frac{1}{4}})$. A corresponding result is obtained if (1) is replaced by the existence of a finite $p$th moment, $p \geqq 4$.
Article DataHistorySubmitted: 04 June 1956Published online: 17 July 2006Publication DataISSN (print): 0040-585XISSN (online): 1095-7219Publisher: Society for Industrial and Applied MathematicsCODEN: tprbau
Article DataHistorySubmitted: 04 June 1956Published online: 17 July 2006Publication DataISSN (print): 0040-585XISSN (online): 1095-7219Publisher: Society for Industrial and Applied MathematicsCODEN: tprbau
Let $(X_j ),j = 1,2, \cdots $, be a sequence of independent random variables with the distribution functions $V_j (x)$. We assume the existence of ${\bf D}X_j = \sigma _j^2 …
Let $(X_j ),j = 1,2, \cdots $, be a sequence of independent random variables with the distribution functions $V_j (x)$. We assume the existence of ${\bf D}X_j = \sigma _j^2 ,s_n^2 = \sum\nolimits_{j = 1}^n {\sigma _j^2 } ,{\bf E}X_j = 0,j = 1,2, \cdots $. We put \[ Z_n = \sum\limits_{j = 1}^n X_j /s_n . \] With the aid of the saddlepoint method of function theory several local limit theorems are derived, in complete analogy to the previously known integral limit theorems for large deviations of H. Cramér [1] and V. Petrov [5]. These authors considered the behavior of the function ${\bf P}\{ Z_n < x\} = F_n (x)$ for $n \to \infty $, where x together with n becomes infinite ("large deviations"). V. Petrov generalized Cramér's theorem from the case of identically distributed $X_j $ to the general case and at the same time improved the remainder term and the growth of x. The present work shows that their method of proof, namely the introduction of a definite transformation of the distribution laws of the $X_j $, was very natural. The present work makes consistent use of the function theoretic possibilities that are given by the assumption that the functions \[ M_j (z) = {\bf E}_{e^{zX_j } } = \int_{ - \infty }^\infty {e^{zy} } dv_j (y) \] are analytic in a strip $| {\operatorname{Re} z} | < A$. Theorem 1.Let conditions A—C be fulfilled. Then for sufficiently largeneach$Z_n $possesses a distribution density$p_{z_n } (x)$. Assume further that$x > 1 $and$x = o(\sqrt n )$for$n \to \infty $. Then one has\[ \frac{{p_{z_n } (x)}}{{\varphi (x)}} = e^{(x/\sqrt n )\lambda _n (x/\sqrt n )} \left[ {1 + O\left( {\frac{x}{{\sqrt n }}} \right)} \right], \]where$\lambda _n (t)$is a power series converging, uniformly inn, for sufficiently small values$| t |$, and$\varphi (x)$is the density o f the normal distribution. For negative x there is a similar relation. For identically distributed $X_j $ the condition C can be considerably weakened. In this case Theorem 2 holds. Also in the case of a lattice-like distribution of the random variables $X_j $ an analogous limit relation holds (Theorem 3).