Xiequan Fan

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Exponential inequalities for martingales with applications

2015-01-01

The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Peña, Pinelis and van de … The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Peña, Pinelis and van de Geer. Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are also provided.

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Hoeffding’s inequality for supermartingales

2012-06-19

Xiequan Fan , Ion Grama , Quansheng Liu

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Cramér large deviation expansions for martingales under Bernstein’s condition

2013-06-28

Xiequan Fan , Ion Grama , Quansheng Liu

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Large deviation exponential inequalities for supermartingales

2012-01-01

Xiequan Fan , Ion Grama , Quansheng Liu

Let $(X_{i}, \mathcal{F}_{i})_{i\geq 1}$ be a sequence of supermartingale differences and let $S_k=\sum_{i=1}^k X_i$. We give an exponential moment condition under which $\mathbb{P}( \max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 … Let $(X_{i}, \mathcal{F}_{i})_{i\geq 1}$ be a sequence of supermartingale differences and let $S_k=\sum_{i=1}^k X_i$. We give an exponential moment condition under which $\mathbb{P}( \max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^{\alpha}\}),$ $n\rightarrow \infty, $ where $\alpha \in $ is given and $C_{1}>0$ is a constant. We also show that the power $\alpha$ is optimal under the given moment condition.

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Deviation inequalities for separately Lipschitz functionals of iterated random functions

2014-08-11

Jérôme Dedecker , Xiequan Fan

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Deviation inequalities for martingales with applications

2016-11-16

Xiequan Fan , Ion Grama , Quansheng Liu

Self-normalized Cramér type moderate deviations for martingales

2019-09-13

Xiequan Fan , Ion Grama , Quansheng Liu +1 more

Let $(X_{i},\mathcal{F}_{i})_{i\geq 1}$ be a sequence of martingale differences. Set $S_{n}=\sum_{i=1}^{n}X_{i}$ and $[S]_{n}=\sum_{i=1}^{n}X_{i}^{2}$. We prove a Cramér type moderate deviation expansion for $\mathbf{P}(S_{n}/\sqrt{[S]_{n}}\geq x)$ as $n\to +\infty $. Our results … Let $(X_{i},\mathcal{F}_{i})_{i\geq 1}$ be a sequence of martingale differences. Set $S_{n}=\sum_{i=1}^{n}X_{i}$ and $[S]_{n}=\sum_{i=1}^{n}X_{i}^{2}$. We prove a Cramér type moderate deviation expansion for $\mathbf{P}(S_{n}/\sqrt{[S]_{n}}\geq x)$ as $n\to +\infty $. Our results partly extend the earlier work of Jing, Shao and Wang (Ann. Probab. 31 (2003) 2167–2215) for independent random variables.

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Exact rates of convergence in some martingale central limit theorems

2018-09-27

Xiequan Fan

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Sharp large deviation results for sums of independent random variables

2015-07-09

Xiequan Fan , Ion Grama , Quansheng Liu

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Cramér moderate deviations for the elephant random walk

2021-02-01

Xiequan Fan , Haijuan Hu , Xiaohui Ma

Abstract We establish some limit theorems for the elephant random walk (ERW), including Berry–Esseen’s bounds, Cramér moderate deviations and local limit theorems. These limit theorems can be regarded as refinements … Abstract We establish some limit theorems for the elephant random walk (ERW), including Berry–Esseen’s bounds, Cramér moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorem for the ERW. Moreover, by these limit theorems, we conclude that the convergence rate of normal approximations and the domain of attraction of normal distribution mainly depend on a memory parameter p which lies between 0 and 3/4.

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Self-normalized Cramér type moderate deviations for stationary sequences and applications

2020-03-09

Xiequan Fan , Ion Grama , Quansheng Liu +1 more

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Cramér’s moderate deviations for martingales with applications

2024-07-31

Xiequan Fan , Qi-Man Shao

Soit (ξi,Fi)i≥1 une suite de différences de martingale. Soient Xn=∑ i=1nξi et ⟨X⟩n=∑i=1nE(ξi2|Fi−1). Nous prouvons les développements de déviation modérée de Cramér pour P(Xn/⟨X⟩n≥x) et P(Xn/ E Xn2≥x) lorsque n→∞. … Soit (ξi,Fi)i≥1 une suite de différences de martingale. Soient Xn=∑ i=1nξi et ⟨X⟩n=∑i=1nE(ξi2|Fi−1). Nous prouvons les développements de déviation modérée de Cramér pour P(Xn/⟨X⟩n≥x) et P(Xn/ E Xn2≥x) lorsque n→∞. Nos résultats étendent le résultat classique de Cramér aux cas des martingales normalisées Xn/⟨X⟩ n et des martingales standardisées Xn/ E Xn2, où les différences de martingale vérifient la condition de Bernstein conditionnelle. Des applications aux marches aléatoires des éléphants et aux processus autorégressifs sont également discutées.

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Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment

2020-10-01

Xiequan Fan , Haijuan Hu , Quansheng Liu

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Exponential inequalities for nonstationary Markov chains

2019-01-01

Pierre Alquier , Paul Doukhan , Xiequan Fan

Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work … Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almost all the results concern stationary time series. This excludes many important applications: for example any series with a periodic behavior is non-stationary. In this paper, we extend the basic tools of Dedecker and Fan (2015) to nonstationary Markov chains. As an application, we provide a Bernstein-type inequality, and we deduce risk bounds for the prediction of periodic autoregressive processes with an unknown period.

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On the Nadaraya–Watson kernel regression estimator for irregularly spaced spatial data

2019-06-27

Mohamed El Machkouri , Xiequan Fan , Lucas Reding

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Martingale inequalities of type Dzhaparidze and van Zanten

2017-04-27

Xiequan Fan , Ion Grama , Quansheng Liu

Freedman's inequality is a supermartingale counterpart to Bennett's inequality. This result shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound … Freedman's inequality is a supermartingale counterpart to Bennett's inequality. This result shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound for the supermartingale difference sequence. Replacing the quadratic characteristic by Hky:=∑i=1k(E(ξi2|Fi−1)+ξi21{|ξi|>y}), Dzhaparidze and van Zanten [On Bernstein-type inequalities for martingales. Stoch Process Appl. 2001;93:109–117] have extended Freedman's inequality to martingales with unbounded differences. In this paper, we prove that Hky can be refined to Gky:=∑i=1k(E(ξi21{ξi≤y}|Fi−1)+ξi21{ξi>y}). Moreover, we also establish two inequalities of type Dzhaparidze and van Zanten. These results extend Sason's inequality [Tightened exponential bounds for discrete-time conditionally symmetric martingales with bounded jumps. Statist Probab Lett. 2013;83:1928–1936] to martingales with possibly unbounded differences and establish the connection between Sason's inequality and De la Peña's inequality [A general class of exponential inequalities for martingales and ratios. Ann Probab. 1999;27(1):537–564]. An application to self-normalized deviations is given.

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Sharp large deviations for sums of bounded from above random variables

2017-09-06

Xiequan Fan

Berry–Esseen Bounds for Self-Normalized Martingales

2017-12-27

Xiequan Fan , Qi-Man Shao

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Tempered fractional multistable motion and tempered multifractional stable motion

2018-07-14

Xiequan Fan , Jacques Lévy Véhel

This work defines two classes of processes, that we term tempered fractional multistable motion and tempered multifractional stable motion . They are extensions of fractional multistable motion and multifractional stable … This work defines two classes of processes, that we term tempered fractional multistable motion and tempered multifractional stable motion . They are extensions of fractional multistable motion and multifractional stable motion, respectively, obtained by adding an exponential tempering to the integrands. We investigate certain basic features of these processes, including scaling property, tail probabilities, absolute moment, sample path properties, pointwise Hölder exponent, Hölder continuity of quasi norm, (strong) localisability and semi-long-range dependence structure. These processes may provide useful models for data that exhibit both dependence and varying local regularity/intensity of jumps.

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Deviation inequalities for separately Lipschitz functionals of composition of random functions

2019-07-05

Jérôme Dedecker , Paul Doukhan , Xiequan Fan

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Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications

2019-10-05

Xiequan Fan , Ion Grama , Quansheng Liu

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On Wasserstein-1 distance in the central limit theorem for elephant random walk

2022-01-01

Xiaohui Ma , Mohamed El Machkouri , Xiequan Fan

Recently, the elephant random walk has attracted a lot of attentions. A wide range of literature is available for the asymptotic behavior of the process, such as the central limit … Recently, the elephant random walk has attracted a lot of attentions. A wide range of literature is available for the asymptotic behavior of the process, such as the central limit theorems, functional limit theorems and the law of iterated logarithm. However, there is not result concerning Wassertein-1 distance for the normal approximations.In this paper, we show that the Wassertein-1 distance in the central limit theorem is totally different when a memory parameter $p$ belongs to one of the three cases $0< p < 1/2,$ $1/2< p<3/4$ and $p=3/4.$

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Sharp large deviations under Bernsteinʼs condition

2013-10-30

Xiequan Fan , Ion Grama , Quansheng Liu

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The missing factor in Bennett's inequality

2012-06-12

Xiequan Fan , Ion Grama , Quansheng Liu

Let Sn be a sum of independent centered random variables satisfying Bernstein's condition with parameter and � 2 be the variance of Sn. Bennett's inequality states that, for any x … Let Sn be a sum of independent centered random variables satisfying Bernstein's condition with parameter and � 2 be the variance of Sn. Bennett's inequality states that, for any x � 0, P(Snx�) � exp � 1 b 2 � , where b = 2x 1+ p 1+2x/� . We give several inequalities which improve this inequality to optimal order, in the spirit of Talagrand's refinement of Hoeffding's inequality. In particular, we sharpen this inequality by adding a missing factor F(x) with exponentially decay rate. The interesting feature of our bound is that it recovers closely the shape of the standard normal tail 1 �(x) for all x � 0, in contrast to Bennett's bound which does not share this property. Also, compared with the classical Cramer large deviations, our inequality has the advantage that it is valid for all x � 0.

Cramér type moderate deviations for self-normalized ψ-mixing sequences

2020-01-28

Xiequan Fan

Rates of Convergence in the Central Limit Theorem for the Elephant Random Walk with Random Step Sizes

2023-09-27

Jérôme Dedecker , Xiequan Fan , Haijuan Hu +1 more

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Multistable Lévy motions and their continuous approximations

2015-01-01

Xiequan Fan , Jacques Lévy Véhel

Multistable Lévy motions are extensions of Lévy motions where the stability index is allowed to vary in time. Several constructions of these processes have been introduced recently, based on Poisson … Multistable Lévy motions are extensions of Lévy motions where the stability index is allowed to vary in time. Several constructions of these processes have been introduced recently, based on Poisson and Ferguson-Klass-LePage series representations and on multistable measures. In this work, we prove a functional central limit theorem for the independent-increments multistable Lévy motion, as well as of integrals with respect to these processes, using weighted sums of independent random variables. This allows us to construct continuous approximations of multistable Lévy motions. In particular, we prove that multistable Lévy motions are stochastic Hölder continuous and strongly localisable.

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Some limit theorems for the elephant random walk

2020-03-29

Xiequan Fan , Haijuan Hu , Xiaohui Ma

On the Wasserstein distance for a martingale central limit theorem

2020-08-08

Xiequan Fan , Xiaohui Ma

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Cramér moderate deviations for a supercritical Galton–Watson process

2022-10-11

Paul Doukhan , Xiequan Fan , Zhiqiang Gao

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Limit theorems and inequalities via martingale methods

2014-01-01

Jean-René Chazottes , Christophe Cuny , Jérôme Dedecker +2 more

In these notes, we first give a brief overwiew of martingales methods, from Paul Lévy (1935) untill now, to explain why these methods have become a central tool in probability, … In these notes, we first give a brief overwiew of martingales methods, from Paul Lévy (1935) untill now, to explain why these methods have become a central tool in probability, statistics and ergodic theory. Next, we present some recent results for/or based on martingales: exponential bounds for super-martingales, concentration inequalities for Lipschitz functionals of dynamical systems, oracle inequalities for the Cox model in a high dimensional setting, and invariance principles for stationary sequences.

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Sharp large deviation probabilities for sums of independent bounded random variables

2012-01-01

Xiequan Fan , Ion Grama , Quansheng Liu

We obtain some optimal inequalities on tail probabilities for sums of independent bounded random variables. Our main result completes an upper bound on tail probabilities due to Talagrand by giving … We obtain some optimal inequalities on tail probabilities for sums of independent bounded random variables. Our main result completes an upper bound on tail probabilities due to Talagrand by giving a one-term asymptotic expansion for large deviations. This result can also be regarded as sharp large deviations of types of Cramér and Bahadur-Ranga Rao.

Gaussian martingale inequality applies to random functions and maxima of empirical processes

2017-01-01

Xiequan Fan

We obtain a Bernstein type Gaussian concentration inequality for martingales. Our inequality improves the Azuma-Hoeffding inequality for moderate deviations $x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron, … We obtain a Bernstein type Gaussian concentration inequality for martingales. Our inequality improves the Azuma-Hoeffding inequality for moderate deviations $x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron, Lugosi and Massart (2000,2003), we show that our result can be applied to the concentration of random functions, Erdös-Rényi random graph, and maxima of empirical processes. Several interesting Gaussian concentration inequalities have been obtained.

Hoeffding's inequality for supermartingales

2011-01-01

Xiequan Fan , Ion Grama , Quansheng Liu

We give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and … We give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.

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About the constant in Talagrand's inequality for sums of bounded random variables

2012-06-12

Xiequan Fan , Ion Grama , Quansheng Liu

A Berry–Esseen bound of order $\protect \frac{1}{\protect \sqrt{n}} $ for martingales

2020-10-02

Songqi Wu , Xiaohui Ma , Hailin Sang +1 more

Renz [13] has established a rate of convergence 1/n in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the methods, developed … Renz [13] has established a rate of convergence 1/n in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the methods, developed by Bolthausen [2] and Grama and Haeusler [6], is applied for obtaining the same convergence rate for a class of more general martingales. An application to linear processes is discussed.

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Spatial-Frequency domain nonlocal total variation for image denoising

2020-01-01

Haijuan Hu , Jacques Froment , Baoyan Wang +1 more

<p style='text-indent:20px;'>Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a … <p style='text-indent:20px;'>Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a large number of denoising methods mixing these two approaches, starting with the nonlocal total variation (NLTV) model. The present article proposes an analysis of the NLTV model for image denoising as well as a number of improvements, the most important of which being to apply the denoising both in the space domain and in the Fourier domain, in order to exploit the complementarity of the representation of image data. A local version obtained by a regionwise implementation followed by an aggregation process, called Local Spatial-Frequency NLTV (L-SFNLTV) model, is finally proposed as a new reference algorithm for image denoising among the family of approaches mixing TV and NL operators. The experiments show the great performance of L-SFNLTV in terms of image quality and of computational speed, comparing with other recently proposed NLTV-related methods.

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Self-normalized Cramér moderate deviations for a supercritical Galton–Watson process

2023-04-24

Xiequan Fan , Qi-Man Shao

Abstract Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for the offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds for the … Abstract Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for the offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.

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A generalization of Cramér large deviations for martingales

2014-09-20

Xiequan Fan , Ion Grama , Quansheng Liu

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Non-uniform Berry–Esseen bounds for martingales with applications to statistical estimation

2017-01-02

Xiequan Fan , Ion Grama , Qiang Liu

We establish non-uniform Berry–Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cramér type large deviations for moderate x's, and are of exponential decay rate as de … We establish non-uniform Berry–Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cramér type large deviations for moderate x's, and are of exponential decay rate as de la Peña's inequality when x→∞. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.

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Exact Berry-Esseen bounds for martingales

2016-08-31

Xiequan Fan

Renz (1996), Ouchti(2005), El Machkouri and Ouchti (2007) and Mourrat (2013) have established the bounds on the rate of convergence in the central limit theorem for discrete time martingales. In … Renz (1996), Ouchti(2005), El Machkouri and Ouchti (2007) and Mourrat (2013) have established the bounds on the rate of convergence in the central limit theorem for discrete time martingales. In the present paper a modification of the methods, developed by Bolthausen (1982) and Grama and Haeusler (2000), is applied for obtaining exact rates of convergence in the central limit theorem for martingales with differences having conditional moments of order $2+\rho, \rho>0$. Our results significantly improve and generalise the bounds of Renz (1996), Ouchti(2005), El Machkouri and Ouchti (2007) and Mourrat (2013). Our results generalise and strengthen the bounds mentioned above. An application to Lipschitz functionals of independent random variables is also given.

McDiarmid's martingale for a class of iterated random functions

2014-02-17

Jérôme Dedecker , Xiequan Fan

We consider a Markov chain X_1, X_2, ..., X_n belonging to a class of iterated random functions, which is one-step contracting with respect to some distance d. If f is … We consider a Markov chain X_1, X_2, ..., X_n belonging to a class of iterated random functions, which is one-step contracting with respect to some distance d. If f is any separately Lipschitz function with respect to d, we use a well known decomposition of S_n=f(X_1, ..., X_n) -E[f(X_1, ..., X_n)]$ into a sum of martingale differences d_k with respect to the natural filtration F_k. We show that each difference d_k is bounded by a random variable eta_k independent of F_{k-1}. Using this very strong property, we obtain a large variety of deviation inequalities for S_n, which are governed by the distribution of the eta_k's. Finally, we give an application of these inequalities to the Wasserstein distance between the empirical measure and the invariant distribution of the chain.

Self-normalized Cramer type moderate deviations for martingales

2017-12-13

Xiequan Fan , Ion Grama , Quansheng Liu +1 more

Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}^n\xi_i $ and $[ S]_n=\sum_{i=1}^n \xi_i^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbf{P}(S_n/\sqrt{[ S]_n} \geq x)$ as $n\to+\infty.$ … Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}^n\xi_i $ and $[ S]_n=\sum_{i=1}^n \xi_i^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbf{P}(S_n/\sqrt{[ S]_n} \geq x)$ as $n\to+\infty.$ Our results partly extend the earlier work of [Jing, Shao and Wang, 2003] for independent random variables.

Bernstein type inequalities for self-normalized martingales with applications

2018-11-26

Xiequan Fan , Shen Wang

For self-normalized martingales with conditionally symmetric differences, de la Peña [A general class of exponential inequalities for martingales and ratios. Ann Probab. 1999;27(1):537–564] established the Gaussian type exponential inequalities. Bercu … For self-normalized martingales with conditionally symmetric differences, de la Peña [A general class of exponential inequalities for martingales and ratios. Ann Probab. 1999;27(1):537–564] established the Gaussian type exponential inequalities. Bercu and Touati [Exponential inequalities for self-normalized martingales with applications. Ann Appl Probab. 2008;18:1848–1869] extended de la Peña's inequalities to martingales with differences heavy on left. In this paper, we establish Bernstein type exponential inequalities for self-normalized martingales with differences bounded from below. Moreover, applications to t-statistics and autoregressive processes are discussed.

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Cramér moderate deviations for the elephant random walk

2020-01-01

Xiequan Fan , Haijuan Hu , Xiaohui Ma

We establish some limit theorems for one-dimensional elephant random walk, including Berry-Esseen bounds, Cram\'{e}r moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the … We establish some limit theorems for one-dimensional elephant random walk, including Berry-Esseen bounds, Cram\'{e}r moderate deviations and local limit theorems. These limit theorems can be regarded as refinements of the central limit theorems for the elephant random walk. Moreover, by these limit theorems, we conclude that the domain of attraction of normal distribution mainly depends on a memory parameter $p$ which lies between $0$ and $3/4.$

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Deviation inequalities for stochastic approximation by averaging

2022-07-15

Xiequan Fan , Pierre Alquier , Paul Doukhan

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Cramér’s large deviation expansion for a supercritical branching process with immigration in a random environment

2021-01-01

Yanqing Wang , Quansheng Liu , Xiequan Fan

Deviation inequalities for martingales with applications to linear regressions and weak invariance principles

2015-01-01

Xiequan Fan

Using changes of probability measure developed by \mbox{Grama} and Haeusler (Stochastic Process.\ Appl., 2000), we obtain two generalizations of the deviation inequalities of Lanzinger and Stadtm\"{u}ller (Stochastic Process.\ Appl., 2000) … Using changes of probability measure developed by \mbox{Grama} and Haeusler (Stochastic Process.\ Appl., 2000), we obtain two generalizations of the deviation inequalities of Lanzinger and Stadtm\"{u}ller (Stochastic Process.\ Appl., 2000) and Fuk and Nagaev (Theory Probab. Appl., 1971) to the case of martingales. Our inequalities recover the best possible decaying rate of independent case. Applications to linear regressions and weak invariance principles for martingales are provided.

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Large deviations for martingales with exponential condition

2011-11-06

Xiequan Fan , Ion Grama , Quansheng Liu

Bernstein type exponential inequalities for martingales

2013-11-25

Xiequan Fan , Ion Grama , Quansheng Liu

The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Pena, Pinelis and van de … The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Pena, Pinelis and van de Geer. Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are provided. In particular, an interesting application of {de la Pena's} inequality to self-normalized deviations is also provided.

Rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales

2025-05-01

Xiequan Fan , Zhonggen Su

Deviation inequalities for the elephant random walk with random step sizes

2025-03-01

Haijuan Hu , Xiequan Fan

Recently, the asymptotic behaviors of the elephant random walk have attracted a lot of attentions. Many interesting properties for the elephant random walk have been established in the past few … Recently, the asymptotic behaviors of the elephant random walk have attracted a lot of attentions. Many interesting properties for the elephant random walk have been established in the past few years, such as strong invariance principles, central limit theorems, functional limit theorems and law of large numbers. In this paper, we give some deviation inequalities for the elephant random walk with random step sizes with various moment conditions. More precisely, we are interested in giving quantitative estimates for the convergence rates in the law of large numbers.

Self-normalized Cram\'er-type Moderate Deviation of Stochastic Gradient Langevin Dynamics

2024-10-29

Hongsheng Dai , Xiequan Fan , Jianya Lu

In this paper, we study the self-normalized Cram\'er-type moderate deviation of the empirical measure of the stochastic gradient Langevin dynamics (SGLD). Consequently, we also derive the Berry-Esseen bound for SGLD. … In this paper, we study the self-normalized Cram\'er-type moderate deviation of the empirical measure of the stochastic gradient Langevin dynamics (SGLD). Consequently, we also derive the Berry-Esseen bound for SGLD. Our approach is by constructing a stochastic differential equation (SDE) to approximate the SGLD and then applying Stein's method as developed in [9,19], to decompose the empirical measure into a martingale difference series sum and a negligible remainder term.

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Pointwise Sharp Moderate Deviations for a Kernel Density Estimator

2024-10-10

Siyu Liu , Xiequan Fan , Haijuan Hu +1 more

Let fn be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random vectors taking values in Rd. With some … Let fn be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random vectors taking values in Rd. With some mild conditions, we establish sharp moderate deviations for the kernel density estimator. This means that we provide an equivalent for the tail probabilities of this estimator.

Deviation inequalities for contractive infinite memory process

2024-08-21

Paul Doukhan , Xiequan Fan

In this paper, we introduce a class of processes that contains many natural examples. The interesting feature of such type processes lays on its infinite memory that allows it to … In this paper, we introduce a class of processes that contains many natural examples. The interesting feature of such type processes lays on its infinite memory that allows it to record a quite ancient history. Then, using the martingale decomposition method, we establish some deviation and moment inequalities for separately Lipschitz functions of such a process, under various moment conditions on some dominating random variables. Our results generalize the Markov models of Dedecker and Fan [Stochastic Process.\ Appl., 2015] and a recent paper by Chazottes et al.\ [Ann.\ Appl.\ Probab., 2023] for the special case of a specific class of infinite memory models with discrete values.

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Cramér’s moderate deviations for martingales with applications

2024-07-31

Xiequan Fan , Qi-Man Shao

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Sharp moderate and large deviations for sample quantiles

2023-10-28

Xiequan Fan

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Rates of Convergence in the Central Limit Theorem for the Elephant Random Walk with Random Step Sizes

2023-09-27

Jérôme Dedecker , Xiequan Fan , Haijuan Hu +1 more

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Self-normalized Cramér moderate deviations for a supercritical Galton–Watson process

2023-04-24

Xiequan Fan , Qi-Man Shao

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Asymptotic expansions in the local limit theorem for a branching Wiener process

2023-04-22

Guantie Deng , Xiequan Fan , Zhiqiang Gao

Rates of convergence in the central limit theorem for the elephant random walk with random step sizes

2023-02-12

Jérôme Dedecker , Xiequan Fan , Haijuan Hu +1 more

In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step … In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically distributed random variables instead of a fixed constant. For this model, we establish the law of the iterated logarithm, the central limit theorem, and we obtain rates of convergence in the central limit theorem with respect to the Kologmorov, Zolotarev and Wasserstein distances. We emphasize that, even in case of the usual elephant random walk, our results concerning the rates of convergence in the central limit theorem are new.

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Sharp moderate and large deviations for sample quantiles

2023-01-01

Xiequan Fan

In this article, we discuss the sharp moderate and large deviations between the quantiles of population and the quantiles of samples. Cram\'{e}r type moderate deviations and Bahadur-Rao type large deviations … In this article, we discuss the sharp moderate and large deviations between the quantiles of population and the quantiles of samples. Cram\'{e}r type moderate deviations and Bahadur-Rao type large deviations are established with some mild conditions. The results refine the moderate and large deviation principles of Xu and Miao [Filomat 2011; 25(2): 197-206].

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Wasserstein-$1$ distance and nonuniform Berry-Esseen bound for a supercritical branching process in a random environment

2023-01-01

Hao Wu , Xiequan Fan , Zhiqiang Gao +1 more

Let $ (Z_{n})_{n\geq 0} $ be a supercritical branching process in an independent and identically distributed random environment. We establish an optimal convergence rate in the Wasserstein-$1$ distance for the … Let $ (Z_{n})_{n\geq 0} $ be a supercritical branching process in an independent and identically distributed random environment. We establish an optimal convergence rate in the Wasserstein-$1$ distance for the process $ (Z_{n})_{n\geq 0} $, which completes a result of Grama et al. [Stochastic Process. Appl., 127(4), 1255-1281, 2017]. Moreover, an exponential nonuniform Berry-Esseen bound is also given. At last, some applications of the main results to the confidence interval estimation for the criticality parameter and the population size $Z_n$ are discussed.

Sharp Moderate and Large Deviations for Sample Quantiles

2023-01-01

Xiequan Fan

In this article, we discuss the sharp moderate and large deviations between the quantiles of population and the quantiles of samples. Cramer type moderate deviations and Bahadur-Rao type large deviations … In this article, we discuss the sharp moderate and large deviations between the quantiles of population and the quantiles of samples. Cramer type moderate deviations and Bahadur-Rao type large deviations are established with some mild conditions. The results refine the moderate and large deviation principles of Xu and Miao [Filomat 2011; 25(2): 197-206].

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Self-normalized Cramér type moderate deviations for martingales and applications

2023-01-01

Xiequan Fan , Qi-Man Shao

Cram\'er's moderate deviations give a quantitative estimate for the relative error of the normal approximation and provide theoretical justifications for many estimator used in statistics. In this paper, we establish … Cram\'er's moderate deviations give a quantitative estimate for the relative error of the normal approximation and provide theoretical justifications for many estimator used in statistics. In this paper, we establish self-normalized Cram\'{e}r type moderate deviations for martingales under some mile conditions. The result extends an earlier work of Fan, Grama, Liu and Shao [Bernoulli, 2019]. Moreover, applications of our result to Student's statistic, stationary martingale difference sequences and branching processes in a random environment are also discussed. In particular, we establish Cram\'{e}r type moderate deviations for Student's $t$-statistic for branching processes in a random environment.

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Rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales

2023-01-01

Xiequan Fan , Zhonggen Su

We give some rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales with differences having finite variances. For the Kolmogorov distances, we present some exact Berry-Esseen … We give some rates of convergence in the distances of Kolmogorov and Wasserstein for standardized martingales with differences having finite variances. For the Kolmogorov distances, we present some exact Berry-Esseen bounds for martingales, which generalizes some Berry-Esseen bounds due to Bolthausen. For the Wasserstein distance, with Stein's method and Lindeberg's telescoping sum argument, the rates of convergence in martingale central limit theorems recover the classical rates for sums of i.i.d.\ random variables, and therefore they are believed to be optimal.

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Cramér moderate deviations for a supercritical Galton–Watson process

2022-10-11

Paul Doukhan , Xiequan Fan , Zhiqiang Gao

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Deviation inequalities for stochastic approximation by averaging

2022-07-15

Xiequan Fan , Pierre Alquier , Paul Doukhan

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On Wasserstein-1 distance in the central limit theorem for elephant random walk

2022-01-01

Xiaohui Ma , Mohamed El Machkouri , Xiequan Fan

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Cramér's moderate deviations for martingales with applications

2022-01-01

Xiequan Fan , Qi-Man Shao

Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $X_n=\sum_{i=1}^n \xi_i $ and $ \langle X \rangle_n=\sum_{i=1}^n \mathbf{E}(\xi_i^2|\mathcal{F}_{i-1}).$ We prove Cram\'er's moderate deviation expansions for $\displaystyle \mathbf{P}(X_n/\sqrt{\langle X\rangle_n} \geq x)$ … Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $X_n=\sum_{i=1}^n \xi_i $ and $ \langle X \rangle_n=\sum_{i=1}^n \mathbf{E}(\xi_i^2|\mathcal{F}_{i-1}).$ We prove Cram\'er's moderate deviation expansions for $\displaystyle \mathbf{P}(X_n/\sqrt{\langle X\rangle_n} \geq x)$ and $\displaystyle \mathbf{P}(X_n/\sqrt{ \mathbf{E}X_n^2} \geq x)$ as $n\to\infty.$ Our results extend the classical Cram\'{e}r result to the cases of normalized martingales $X_n/\sqrt{\langle X\rangle_n}$ and standardized martingales $X_n/\sqrt{ \mathbf{E}X_n^2}$, with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed.

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Comparison on the criticality parameters for two supercritical branching processes in random environments

2022-01-01

Xiequan Fan , Haijuan Hu , Hao Wu +1 more

Let $\{Z_{1,n} , n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$ be two supercritical branching processes in different random environments, with criticality parameters $\mu_1$ and $\mu_2$ respectively. It is known that $\frac{1}{n} … Let $\{Z_{1,n} , n\geq 0\}$ and $\{Z_{2,n}, n\geq 0\}$ be two supercritical branching processes in different random environments, with criticality parameters $\mu_1$ and $\mu_2$ respectively. It is known that $\frac{1}{n} \ln Z_{1,n} \rightarrow \mu_1$ and $\frac{1}{m} \ln Z_{2,m} \rightarrow \mu_2$ in probability as $m, n \rightarrow \infty.$ In this paper, we are interested in the comparison on the two criticality parameters. To this end, we prove a non-uniform Berry-Esseen's bound and Cram\'{e}r's moderate deviations for $\frac{1}{n} \ln Z_{1,n} - \frac{1}{m} \ln Z_{2,m}$ as $m, n \rightarrow \infty.$ An application is also given for constructing confidence interval for $\mu_1-\mu_2$.

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Cramér-type moderate deviations for Euler-Maruyama scheme for SDE

2022-01-01

Xiequan Fan , Haijuan Hu , Lihu Xu

In this paper, we establish normalized and self-normalized Cram\'er-type moderate deviations for Euler-Maruyama scheme for SDE. As a consequence of our results, Berry-Esseen's bounds and moderate deviation principles are also … In this paper, we establish normalized and self-normalized Cram\'er-type moderate deviations for Euler-Maruyama scheme for SDE. As a consequence of our results, Berry-Esseen's bounds and moderate deviation principles are also obtained. Our normalized Cram\'er-type moderate deviations refines the recent work of [Lu, J., Tan, Y., Xu, L., 2022. Central limit theorem and self-normalized Cram\'er-type moderate deviation for Euler-Maruyama scheme. Bernoulli 28(2): 937--964].

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On Wassertein-1 distance in the central limit theorem for elephant random walk.

2021-11-21

Xiaohui Ma , Mohamed El Machkouri , Xiequan Fan

A Review on some weak dependence conditions

2021-08-25

Benjamin Bobbia , Paul Doukhan , Xiequan Fan

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Cramér moderate deviations for the elephant random walk

2021-02-01

Xiequan Fan , Haijuan Hu , Xiaohui Ma

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Cramér moderate deviations for a supercritical Galton-Watson process

2021-01-01

Paul Doukhan , Xiequan Fan , Zhiqiang Gao

Let $(Z_n)_{n\geq0}$ be a supercritical Galton-Watson process. The Lotka-Nagaev estimator $Z_{n+1}/Z_n$ is a common estimator for the offspring mean.In this paper, we establish some Cram\'{e}r moderate deviation results for the … Let $(Z_n)_{n\geq0}$ be a supercritical Galton-Watson process. The Lotka-Nagaev estimator $Z_{n+1}/Z_n$ is a common estimator for the offspring mean.In this paper, we establish some Cram\'{e}r moderate deviation results for the Lotka-Nagaev estimator via a martingale method. Applications to construction of confidence intervals are also given.

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Cramér’s large deviation expansion for a supercritical branching process with immigration in a random environment

2021-01-01

Yanqing Wang , Quansheng Liu , Xiequan Fan

Deviation inequalities for stochastic approximation by averaging

2021-01-01

Xiequan Fan , Pierre Alquier , Paul Doukhan

We introduce a class of Markov chains, that contains the model of stochastic approximation by averaging and non-averaging. Using martingale approximation method, we establish various deviation inequalities for separately Lipschitz … We introduce a class of Markov chains, that contains the model of stochastic approximation by averaging and non-averaging. Using martingale approximation method, we establish various deviation inequalities for separately Lipschitz functions of such a chain, with different moment conditions on some dominating random variables of martingale differences.Finally, we apply these inequalities to the stochastic approximation by averaging and empirical risk minimisation.

A Berry–Esseen bound of order $\protect \frac{1}{\protect \sqrt{n}} $ for martingales

2020-10-02

Songqi Wu , Xiaohui Ma , Hailin Sang +1 more

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Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment

2020-10-01

Xiequan Fan , Haijuan Hu , Quansheng Liu

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On the Wasserstein distance for a martingale central limit theorem

2020-08-08

Xiequan Fan , Xiaohui Ma

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On the Wassertein distance for a martingale central limit theorem

2020-05-02

Xiequan Fan , Xiaohui Ma

We prove an upper bound on the Wassertein distance between normalized martingales and the standard normal random variable, which extends a result of R\"ollin [Statist. Probabil. Lett. 138 (2018) 171-176]. … We prove an upper bound on the Wassertein distance between normalized martingales and the standard normal random variable, which extends a result of R\"ollin [Statist. Probabil. Lett. 138 (2018) 171-176]. The proof is based on a method of Bolthausen [Ann. Probab. 10 (1982) 672-688].

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Some limit theorems for the elephant random walk

2020-03-29

Xiequan Fan , Haijuan Hu , Xiaohui Ma

Self-normalized Cramér type moderate deviations for stationary sequences and applications

2020-03-09

Xiequan Fan , Ion Grama , Quansheng Liu +1 more

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Cramér type moderate deviations for self-normalized ψ-mixing sequences

2020-01-28

Xiequan Fan

A Berry-Esseen bound of order $ 1/\sqrt{n} $ for martingales

2020-01-01

Songqi Wu , Xiaohui Ma , Hailin Sang +1 more

Renz (Ann. Probab. 1996) has established a rate of convergence $1/\sqrt{n}$ in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the … Renz (Ann. Probab. 1996) has established a rate of convergence $1/\sqrt{n}$ in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the methods, developed by Bolthausen (Ann. Probab. 1982) and Grama and Haeusler (Stochastic Process. Appl. 2000), is applied for obtaining the same convergence rate for a class of more general martingales. An application to linear processes is discussed.

Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment

2020-01-01

Xiequan Fan , Haijuan Hu , Quansheng Liu

Let $\{Z_n, n\geq 0\}$ be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for $\ln (Z_{n+n_0}/Z_{n_0})$ % under … Let $\{Z_n, n\geq 0\}$ be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for $\ln (Z_{n+n_0}/Z_{n_0})$ % under the annealed law, uniformly in $n_0 \in \mathbb{N}$, which extend the corresponding results by Grama et al. (Stochastic Process.\ Appl. 2017) established for $n_0=0$. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. (2017) are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of $\ln(Z_{n+n_0}/Z_{n_0})$ and $n$.

Spatial-Frequency domain nonlocal total variation for image denoising

2020-01-01

Haijuan Hu , Jacques Froment , Baoyan Wang +1 more

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Cramér moderate deviations for the elephant random walk

2020-01-01

Xiequan Fan , Haijuan Hu , Xiaohui Ma

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Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications

2019-10-05

Xiequan Fan , Ion Grama , Quansheng Liu

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Self-normalized Cramér type moderate deviations for martingales

2019-09-13

Xiequan Fan , Ion Grama , Quansheng Liu +1 more

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Deviation inequalities for separately Lipschitz functionals of composition of random functions

2019-07-05

Jérôme Dedecker , Paul Doukhan , Xiequan Fan

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Deviation inequalities for separately Lipschitz functionals of composition of random functions

2019-07-03

Jérôme Dedecker , Paul Doukhan , Xiequan Fan

We consider a class of non-homogeneous Markov chains, that contains many natural examples. Next, using martingale methods, we establish some deviation and moment inequalities for separately Lipschitz functions of such … We consider a class of non-homogeneous Markov chains, that contains many natural examples. Next, using martingale methods, we establish some deviation and moment inequalities for separately Lipschitz functions of such a chain, under moment conditions on some dominating random variables.

On the Nadaraya–Watson kernel regression estimator for irregularly spaced spatial data

2019-06-27

Mohamed El Machkouri , Xiequan Fan , Lucas Reding

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Cramér-type moderate deviations for stationary sequences of bounded random variables

2019-05-01

Xiequan Fan

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Cramér type moderate deviations for stationary sequences of bounded random variables

2019-01-01

Xiequan Fan

We derive Cram\'{e}r type moderate deviations for stationary sequences of bounded random variables. Our results imply the moderate deviation principles and a Berry-Esseen bound. Applications to quantile coupling inequalities, functions … We derive Cram\'{e}r type moderate deviations for stationary sequences of bounded random variables. Our results imply the moderate deviation principles and a Berry-Esseen bound. Applications to quantile coupling inequalities, functions of $\phi$-mixing sequences, and contracting Markov chains are discussed.

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Large deviation inequalities for martingales in Banach spaces

2019-01-01

Xiequan Fan , Davide Giraudo

Let $(X_i, \mathcal{F}_i)_{i\geq1}$ be a martingale difference sequence in a smooth Banach space. Let $S_n=\sum_{i=1}^nX_i, n\geq 1,$ be the partial sums of $(X_i, \mathcal{F}_i)_{i\geq 1}$. We give upper bounds on … Let $(X_i, \mathcal{F}_i)_{i\geq1}$ be a martingale difference sequence in a smooth Banach space. Let $S_n=\sum_{i=1}^nX_i, n\geq 1,$ be the partial sums of $(X_i, \mathcal{F}_i)_{i\geq 1}$. We give upper bounds on the quantity $\mathbb{P}\left(\max_{1\leq k\leq n}\lVert S_k\rVert>nx\right)$ in terms of $ n\geq 1$ and $x>0$ in two different situations: when the martingale differences have uniformly bounded exponential moments and when the decay of the tail of the increments is polynomial.

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Spatial-Frequency Domain Nonlocal Total Variation for Image Denoising

2019-01-01

Haijuan Hu , Jacques Froment , Baoyan Wang +1 more

Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a large … Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a large number of denoising methods mixing these two approaches, starting with the nonlocal total variation (NLTV) model. The present article proposes an analysis of the NLTV model for image denoising as well as a number of improvements, the most important of which being to apply the denoising both in the space domain and in the Fourier domain, in order to exploit the complementarity of the representation of image data in both domains. A local version obtained by a regionwise implementation followed by an aggregation process, called Local Spatial-Frequency NLTV (L- SFNLTV) model, is finally proposed as a new reference algorithm for image denoising among the family of approaches mixing TV and NL operators. The experiments show the great performance of L-SFNLTV, both in terms of image quality and of computational speed, comparing with other recently proposed NLTV-related methods.

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Exponential inequalities for nonstationary Markov chains

2019-01-01

Pierre Alquier , Paul Doukhan , Xiequan Fan

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Deviation inequalities for separately Lipschitz functionals of composition of random functions

2019-01-01

Jérôme Dedecker , Paul Doukhan , Xiequan Fan

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Coauthor	Papers Together
Quansheng Liu	30
Ion Grama	28
Paul Doukhan	11
Xiaohui Ma	9
Qi-Man Shao	9
Jérôme Dedecker	8
Haijuan Hu	7
Haijuan Hu	4
Zhiqiang Gao	4
Jacques Lévy Véhel	3
Pierre Alquier	3
Mohamed El Machkouri	3
Sarah Lemler	2
Yinna Ye	2
Baoyan Wang	2
Jean-René Chazottes	2
Florence Merlevède	2
Jacques Froment	2
Hailin Sang	2
Haijuan Hu	2
Songqi Wu	2
Zhonggen Su	2
Christophe Cuny	2
Lucas Reding	1
J Dedecker	1
Siyu Liu	1
Shen Wang	1
Shen Wang	1
Qiang Liu	1
Hao Wu	1
Hao Wu	1
Lihu Xu	1
Hongsheng Dai	1
Yanqing Wang	1
Jianya Lu	1
Davide Giraudo	1
Guantie Deng	1
Benjamin Bobbia	1

A General Class of Exponential Inequalities for Martingales and Ratios

1999-01-01

Víctor Peña

In this paper we introduce a technique for obtaining exponential inequalities, with particular emphasis placed on results involving ratios. Our main applications consist of approximations to the tail probability of … In this paper we introduce a technique for obtaining exponential inequalities, with particular emphasis placed on results involving ratios. Our main applications consist of approximations to the tail probability of the ratio of a martingale over its conditional variance (or its quadratic variation for continuous martingales). We provide examples that strictly extend several of the classical exponential inequalities for sums of independent random variables and martingales. The spirit of this application is that, when going from results for sums of independent random variables to martingales, one should replace the variance by the conditional variance and the exponential of a function of the variance by the expectation of the exponential of the same function of the conditional variance. The decoupling inequalities used to attain our goal are of independent interest. They include a new exponential decoupling inequality with constraints and a sharp inequality for the probability of the intersection of a fixed number of dependent sets. Finally, we also present an exponential inequality that does not require any integrability conditions involving the ratio of the sum of conditionally symmetric variables to its sum of squares.

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Cramér large deviation expansions for martingales under Bernstein’s condition

2013-06-28

Xiequan Fan , Ion Grama , Quansheng Liu

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Exact Convergence Rates in Some Martingale Central Limit Theorems

1982-08-01

Erwin Bolthausen

Convergence rates are derived in central limit theorems for martingale difference arrays. The rates depend heavily on the behavior of the conditional variances and on moment conditions. It is also … Convergence rates are derived in central limit theorems for martingale difference arrays. The rates depend heavily on the behavior of the conditional variances and on moment conditions. It is also shown that the rates which are obtained are the exact ones under the stated conditions.

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Exponential inequalities for self-normalized martingales with applications

2008-10-01

Bernard Bercu , Abderrahmen Touati

We propose several exponential inequalities for self-normalized martingales similar to those established by De la Peña. The keystone is the introduction of a new notion of random variable heavy on … We propose several exponential inequalities for self-normalized martingales similar to those established by De la Peña. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching processes are also provided.

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Hoeffding’s inequality for supermartingales

2012-06-19

Xiequan Fan , Ion Grama , Quansheng Liu

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Probability Inequalities for the Sum of Independent Random Variables

1962-03-01

G. G. Bennett

Abstract This paper proves a number of inequalities which improve on existing upper limits to the probability distribution of the sum of independent random variables. The inequalities presented require knowledge … Abstract This paper proves a number of inequalities which improve on existing upper limits to the probability distribution of the sum of independent random variables. The inequalities presented require knowledge only of the variance of the sum and the means and bounds of the component random variables. They are applicable when the number of component random variables is small and/or have different distributions. Figures show the improvement on existing inequalities.

Large deviations for martingales

2001-11-01

Emmanuel Lesigne , Dalibor Volný

Large Deviations of Sums of Independent Random Variables

1979-10-01

S. V. Nagaev

This paper deals with numerous variants of bounds for probabilities of large deviations of sums of independent random variables in terms of ordinary and generalized moments of individual summands. A … This paper deals with numerous variants of bounds for probabilities of large deviations of sums of independent random variables in terms of ordinary and generalized moments of individual summands. A great deal of attention is devoted to the study of the precision of these bounds. In this connection comparisons are made with precise asymptotic results. At the end of the paper various applications of the bounds for probabilities of large deviations to the strong law of large numbers, the central limit theorem and to certain other problems are discussed.

On Tail Probabilities for Martingales

1975-02-01

David A. Freedman

Watch a martingale with uniformly bounded increments until it first crosses the horizontal line of height $a$. The sum of the conditional variances of the increments given the past, up … Watch a martingale with uniformly bounded increments until it first crosses the horizontal line of height $a$. The sum of the conditional variances of the increments given the past, up to the crossing, is an intrinsic measure of the crossing time. Simple and fairly sharp upper and lower bounds are given for the Laplace transform of this crossing time, which show that the distribution is virtually the same as that for the crossing time of Brownian motion, even in the tail. The argument can be adapted to extend inequalities of Bernstein and Kolmogorov to the dependent case, proving the law of the iterated logarithm for martingales. The argument can also be adapted to prove Levy's central limit theorem for martingales. The results can be extended to martingales whose increments satisfy a growth condition.

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Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes

1995-10-01

Sara van de Geer

We obtain an exponential probability inequality for martingales and a uniform probability inequality for the process $\int g dN$, where $N$ is a counting process and where $g$ varies within … We obtain an exponential probability inequality for martingales and a uniform probability inequality for the process $\int g dN$, where $N$ is a counting process and where $g$ varies within a class of predictable functions $\mathscr{G}$. For the latter, we use techniques from empirical process theory. The uniform inequality is shown to hold under certain entropy conditions on $\mathscr{G}$. As an application, we consider rates of convergence for (nonparametric) maximum likelihood estimators for counting processes. A similar result for discrete time observations is also presented.

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Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions

2008-03-26

Emmanuel Rio

Exact convergence rates in the central limit theorem for a class of martingales

2007-11-01

Mohamed El Machkouri , Lahcen Ouchti

We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the … We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and, for stationary sequences, the rate n−1/2log n is reached.

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Exponential inequalities for martingales with applications

2015-01-01

Xiequan Fan , Ion Grama , Quansheng Liu

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Large deviations for martingales with some applications

1995-01-01

Alfredas Račkauskas

On moderate deviations for martingales

1997-01-01

Ion Grama

Let $X^n = (X_t^n, \mathscr{F}_t^n)_{0 \leq t \leq 1}$ be square integrable martingales with the quadratic characteristics $\langle X^n \rangle, n = 1, 2, \dots$. We prove that the large … Let $X^n = (X_t^n, \mathscr{F}_t^n)_{0 \leq t \leq 1}$ be square integrable martingales with the quadratic characteristics $\langle X^n \rangle, n = 1, 2, \dots$. We prove that the large deviations relation $P(X_1^n \geq r)/(1 - \Phi (r)) \to 1$ holds true for $r$ growing to infinity with some rate depending on $L_{2\delta}^n = E \sum_{0\leq t\leq 1}| \Delta X_t^n |^{2 + 2 \delta}$ and $N_{2 \delta}^n = E | \langle X^n \rangle_1 - 1|^{1 + \delta}$, where $\delta > 0$ and $L_{2 \delta}^n \to 0$, N_{2 \delta}^n \to 0$ as $n \to \infty$. The exact bound for the remainder is also obtained.

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Exact rates of convergence in some martingale central limit theorems

2018-09-27

Xiequan Fan

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On McDiarmid's concentration inequality

2013-01-01

Emmanuel Rio

In this paper we improve the rate function in the McDiarmid concentration inequality for separately Lipschitz functions of independent random variables. In particular the rate function tends to infinity at … In this paper we improve the rate function in the McDiarmid concentration inequality for separately Lipschitz functions of independent random variables. In particular the rate function tends to infinity at the boundary. We also prove that in some cases the usual normalization factor is not adequate and may be improved.

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Sur un nouveau théorème-limite de la théorie des probabilités

1994-01-01

Harald Cramér

Limit Theorems for Large Deviations

1991-01-01

L. Saulis , V. Statulevičius

Large deviation exponential inequalities for supermartingales

2012-01-01

Xiequan Fan , Ion Grama , Quansheng Liu

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Optimum Bounds for the Distributions of Martingales in Banach Spaces

1994-10-01

Iosif Pinelis

A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is … A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on infinite-dimensional martingales, most of the results seem to be new even for one-dimensional martingales. Moreover, the bounds on moments of the Rosenthal-Burkholder type seem to be to a certain extent new even for sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.

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Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment

2009-05-14

Quansheng Liu , Frédérique Watbled

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On the method of bounded differences

1989-08-03

Colin McDiarmid

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On the rate of convergence in the martingale central limit theorem

2013-03-13

Jean-Christophe Mourrat

Consider a discrete-time martingale, and let $V^2$ be its normalized quadratic variation. As $V^2$ approaches 1, and provided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale … Consider a discrete-time martingale, and let $V^2$ be its normalized quadratic variation. As $V^2$ approaches 1, and provided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian distribution. For any $p\geq 1$, (Ann. Probab. 16 (1988) 275-299) gave a bound on the rate of convergence in this central limit theorem that is the sum of two terms, say $A_p+B_p$, where up to a constant, $A_p={\|V^2-1\|}_p^{p/(2p+1)}$. Here we discuss the optimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, (Ann. Probab. 10 (1982) 672-688) sketched a strategy to prove optimality for $p=1$. Here we extend this strategy to any $p\geq 1$, thereby justifying the optimality of the term $A_p$. As a necessary step, we also provide a new bound on the rate of convergence in the central limit theorem for martingales with bounded increments that improves on the term $B_p$, generalizing another result of (Ann. Probab. 10 (1982) 672-688).

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On Bernstein-type inequalities for martingales

2001-05-01

K. Dzhaparidze , J. H. van Zanten

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Self-normalized large deviations

1997-01-01

Qi-Man Shao

Let ${X, X_n, n \geq 1}$ be a sequence of independent and identically distributed random variables. The classical Cramér-Chernoff large deviation states that $\lim_{n\to\infty} n^{-1} \ln P((\sum_{i=1}^n X_i)/n \geq x) … Let ${X, X_n, n \geq 1}$ be a sequence of independent and identically distributed random variables. The classical Cramér-Chernoff large deviation states that $\lim_{n\to\infty} n^{-1} \ln P((\sum_{i=1}^n X_i)/n \geq x) = \ln \rho (x)$ if and only if the moment generating function of $X$ is finite in a right neighborhood of zero. This paper uses $n^{(p-1)/p} V_{n,p} = n^{(p-1)/p}(\sum_{i=1}^n |X_i|^p)^{1/p} (p > 1)$ as the normalizing constant to establish a self-normalized large deviation without any moment conditions. A self-normalized moderate deviation, that is, the asymptotic probability of $P(S_n/V_{n,p} \geq x_n) for $x_n = o(n^{(p-1)/p})$, is also found for any $X$ in the domain of attraction of a normal or stable law. As a consequence, a precise constant in the self-normalized law of the iterated logarithm of Griffin and Kuelbs is obtained. Applications to the limit distribution of self-normalized sums, the asymptotic probability of the $t$-statistic as well as to the Erdös-Rényi-Shepp law of large numbers are also discussed.

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Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk

2004-10-13

Gunter M. Schütz , Steffen Trimper

We consider a discrete-time random walk where the random increment at time step t depends on the full history of the process. We calculate exactly the mean and variance of … We consider a discrete-time random walk where the random increment at time step t depends on the full history of the process. We calculate exactly the mean and variance of the position and discuss its dependence on the initial condition and on the memory parameter p . At a critical value p((1) )(c ) =1/2 where memory effects vanish there is a transition from a weakly localized regime [where the walker (elephant) returns to its starting point] to an escape regime. Inside the escape regime there is a second critical value where the random walk becomes superdiffusive. The probability distribution is shown to be governed by a non-Markovian Fokker-Planck equation with hopping rates that depend both on time and on the starting position of the walk. On large scales the memory organizes itself into an effective harmonic oscillator potential for the random walker with a time-dependent spring constant k=(2p-1)/t . The solution of this problem is a Gaussian distribution with time-dependent mean and variance which both depend on the initiation of the process.

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Sharp Large Deviations for the Ornstein--Uhlenbeck Process

2002-01-01

Bernard Bercu , Alain Rouault

We establish sharp large deviation principles for well-known random variables associated with the Ornstein--Uhlenbeck process, such as the energy, the maximum likelihood estimator of the drift parameter, and the log-likelihood … We establish sharp large deviation principles for well-known random variables associated with the Ornstein--Uhlenbeck process, such as the energy, the maximum likelihood estimator of the drift parameter, and the log-likelihood ratio.

Deviation inequalities for separately Lipschitz functionals of iterated random functions

2014-08-11

Jérôme Dedecker , Xiequan Fan

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On Deviations of the Sample Mean

1960-12-01

R. R. Bahadur , Rema Rao

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Self-normalized Cramér type moderate deviations for martingales

2019-09-13

Xiequan Fan , Ion Grama , Quansheng Liu +1 more

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Elephant random walks and their connection to Pólya-type urns

2016-11-21

Erich Baur , Jean Bertoin

In this paper, we explain the connection between the elephant random walk (ERW) and an urn model \`a la P\'olya and derive functional limit theorems for the former. The ERW … In this paper, we explain the connection between the elephant random walk (ERW) and an urn model \`a la P\'olya and derive functional limit theorems for the former. The ERW model was introduced in [Phys. Rev. E 70, 045101 (2004)] to study memory effects in a highly non-Markovian setting. More specifically, the ERW is a one-dimensional discrete-time random walk with a complete memory of its past. The influence of the memory is measured in terms of a memory parameter $p$ between zero and one. In the past years, a considerable effort has been undertaken to understand the large-scale behavior of the ERW, depending on the choice of $p$. Here, we use known results on urns to explicitly solve the ERW in all memory regimes. The method works as well for ERWs in higher dimensions and is widely applicable to related models.

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Central limit theorem and related results for the elephant random walk

2017-05-01

Cristian F. Coletti , Renato Jacob Gava , Gunter M. Schütz

We study the so-called elephant random walk (ERW) which is a non-Markovian discrete-time random walk on ℤ with unbounded memory which exhibits a phase transition from a diffusive to superdiffusive … We study the so-called elephant random walk (ERW) which is a non-Markovian discrete-time random walk on ℤ with unbounded memory which exhibits a phase transition from a diffusive to superdiffusive behavior. We prove a law of large numbers and a central limit theorem. Remarkably the central limit theorem applies not only to the diffusive regime but also to the phase transition point which is superdiffusive. Inside the superdiffusive regime, the ERW converges to a non-degenerate random variable which is not normal. We also obtain explicit expressions for the correlations of increments of the ERW.

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Moderate deviations for martingales and mixing random processes

1996-02-01

Fuqing Gao

A Bernstein type inequality and moderate deviations for weakly dependent sequences

2010-06-08

Florence Merlevède , Magda Peligrad , Emmanuel Rio

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Sharp large deviation results for sums of independent random variables

2015-07-09

Xiequan Fan , Ion Grama , Quansheng Liu

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Cramér moderate deviations for the elephant random walk

2021-02-01

Xiequan Fan , Haijuan Hu , Xiaohui Ma

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Sharp large deviations in nonparametric estimation

2006-04-01

Cyrille Joutard

Large deviation results for the kernel density estimator and the kernel regression estimator have been given by Louani [Louani, D., 1998, Large deviations limit theorems for the kernel density estimator. … Large deviation results for the kernel density estimator and the kernel regression estimator have been given by Louani [Louani, D., 1998, Large deviations limit theorems for the kernel density estimator. Scandinavian Journal of Statistics, 25, 243–253; Louani, D., 1999, Some large deviations limit theorems in conditional nonparametric statistics. Statistics, 33, 171–196]. We complete these works by establishing sharp large deviation results for the two estimators. This means that we study precisely the tail probabilities of the estimators. We distinguish two cases depending on the support of the kernel. To prove the results, we need an Edgeworth expansion obtained from a version of Cramer’s condition.

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The missing factor in Hoeffding's inequalities

1995-01-01

Michel Talagrand

Sums of Independent Random Variables

1975-01-01

V. V. Petrov

Iterated random functions

2017-02-24

Persi Diaconis , David A. Freedman

A bstract.Iterated random functions are used to draw pictures or simulate large Ising models, among other applications.They offer a method for studying the steady state distribution of a Markov chain, … A bstract.Iterated random functions are used to draw pictures or simulate large Ising models, among other applications.They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples.The present paper surveys the field and presents some new examples.There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. Introduction.The applied probability literature is nowadays quite daunting.Even relatively simple topics, like Markov chains, have generated enormous complexity.This paper describes a simple idea that helps to unify many arguments in Markov chains, simulation algorithms, control theory, queuing, and other branches of applied probability.The idea is that Markov chains can be constructed by iterating random functions on the state space S.More specifically, there is a family {f θ : θ ∈ Θ} of functions that map S into itself, and a probability distribution µ on Θ.If the chain is at x ∈ S, it moves by choosing θ at random from µ, and going to f θ (x).For now, µ does not depend on x.The process can be written aswhere θ 1 , θ 2 , . . .are independent draws from µ.The Markov property is clear: given the present position of the chain, the conditional distribution of the future does not depend on the past.

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Self-normalized Cramér type moderate deviations for stationary sequences and applications

2020-03-09

Xiequan Fan , Ion Grama , Quansheng Liu +1 more

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Deviation probability bound for martingales with applications to statistical estimation

2000-02-01

R. Liptser , Vladimir Spokoiny

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Self-normalized limit theorems: A survey

2013-01-01

Qi-Man Shao , Qiying Wang

Let $X_{1},X_{2},\ldots,$ be independent random variables with $EX_{i}=0$ and write $S_{n}=\sum_{i=1}^{n}X_{i}$ and $V_{n}^{2}=\sum_{i=1}^{n}X_{i}^{2}$. This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute … Let $X_{1},X_{2},\ldots,$ be independent random variables with $EX_{i}=0$ and write $S_{n}=\sum_{i=1}^{n}X_{i}$ and $V_{n}^{2}=\sum_{i=1}^{n}X_{i}^{2}$. This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum $S_{n}/V_{n}$. Other self-normalized limit theorems are also briefly discussed.

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Moments, moderate and large deviations for a branching process in a random environment

2011-09-22

Chunmao Huang , Quansheng Liu

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Deviation inequalities for martingales with applications

2016-11-16

Xiequan Fan , Ion Grama , Quansheng Liu

On Hoeffding’s inequalities

2004-04-01

V. Bentkus

In a celebrated work by Hoeffding [ J. Amer. Statist. Assoc. 58 (1963) 13–30], several inequalities for tail probabilities of sums Mn={X}1+⋯+{X}n of bounded independent random variables Xj were proved. … In a celebrated work by Hoeffding [ J. Amer. Statist. Assoc. 58 (1963) 13–30], several inequalities for tail probabilities of sums Mn={X}1+⋯+{X}n of bounded independent random variables Xj were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Études Sci. Publ. Math. 81 (1995a) 73–205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257–314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that ℙ{Mn≥x}≤cℙ{Sn≥x}, where c is an absolute constant and Sn={ɛ}1+⋯+{ɛ}n is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x∈ℝ where the survival function x↦ℙ{Sn≥x} has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate ℙ{Sn≥x} special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. The inequalities have applications to measure concentration, leading to results of the type where, up to an absolute constant, the measure concentration is dominated by the concentration in a simplest appropriate model, such results will be considered elsewhere.

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Concentration Inequalities for Sums and Martingales

2015-01-01

Bernard Bercu , Bernard Delyon , Emmanuel Rio

A Bennett concentration inequality and its application to suprema of empirical processes

2002-01-01

Olivier Bousquet

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A Nonuniform Bound on the Rate of Convergence in the Martingale Central Limit Theorem

1988-10-01

Erich Haeusler , Konrad Joos

The main result of the present paper is a sharp nonuniform bound on the rate of convergence to normality in the central limit theorem for martingales having finite moments of … The main result of the present paper is a sharp nonuniform bound on the rate of convergence to normality in the central limit theorem for martingales having finite moments of order $2 + 2\delta$ for some $0 < \delta < \infty$. A nonuniform bound on the rate for convergence to mixtures of normal distributions is obtained as a consequence.

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