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Xiequan Fan

Publications and source records attributed to Xiequan Fan.

At least 19 recordsLinked to original sources
arXiv2025

Deviation inequalities for contractive infinite memory processes

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. An application to stochastic gradient Langevin dynamic algorithm is also discussed.

math.PR↗
arXiv2024

Self-normalized Cramér-type Moderate Deviation of Stochastic Gradient Langevin Dynamics

In this paper, we study the self-normalized Cramér-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.

math.PR↗
arXiv2023

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

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.

math.PR↗
arXiv2023

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

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.

math.PR↗
arXiv2023

Self-normalized Cramér type moderate deviations for martingales and applications

Cramér'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 Cramér 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 Cramér type moderate deviations for Student's $t$-statistic for branching processes in a random environment.

math.PR↗
arXiv2023

Sharp moderate and large deviations for sample quantiles

In this article, we discuss the sharp moderate and large deviations between the quantiles of population and the quantiles of samples. Cramér 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].

math.ST↗
arXiv2023

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

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 $μ_1$ and $μ_2$ respectively. It is known that $\frac{1}{n} \ln Z_{1,n} \rightarrow μ_1$ and $\frac{1}{m} \ln Z_{2,m} \rightarrow μ_2$ 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 Cramér'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 $μ_1-μ_2$.

math.PR↗
arXiv2023

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

In this paper, we establish normalized and self-normalized Cramér-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 Cramér-type moderate deviations refines the recent work of [Lu, J., Tan, Y., Xu, L., 2022. Central limit theorem and self-normalized Cramér-type moderate deviation for Euler-Maruyama scheme. Bernoulli 28(2): 937--964].

math.PR↗
arXiv2023

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

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.

math.PR↗
arXiv2022

Cramér moderate deviations for a supercritical Galton-Watson process

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 Cramér moderate deviation results for the Lotka-Nagaev estimator via a martingale method. Applications to construction of confidence intervals are also given.

math.PR↗
arXiv2022

Cramér's moderate deviations for martingales with applications

Let $(ξ_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $X_n=\sum_{i=1}^n ξ_i $ and $ \langle X \rangle_n=\sum_{i=1}^n \mathbf{E}(ξ_i^2|\mathcal{F}_{i-1}).$ We prove Cramér'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 Cramér 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.

math.PR↗
arXiv2022

Deviation inequalities for stochastic approximation by averaging

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.

math.PR↗
arXiv2022

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

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 $p=3/4.$

math.PR↗
arXiv2021

Cramér moderate deviations for the elephant random walk

We establish some limit theorems for one-dimensional elephant random walk, including Berry-Esseen bounds, Cramér 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 $3/4.$

math.PR↗
arXiv2020

On the Wassertein distance for a martingale central limit theorem

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

math.PR↗
arXiv2020

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

Let $(X _i)_{i\geq1}$ be a stationary sequence. Denote $m=\lfloor n^α\rfloor, 0< α< 1,$ and $ k=\lfloor n/m \rfloor,$ where $\lfloor a \rfloor$ stands for the integer part of $a.$ Set $S_{j}^\circ = \sum_{i=1}^m X_{m(j-1)+i}, 1\leq j \leq k,$ and $ (V_k^\circ)^2 = \sum_{j=1}^k (S_{j}^\circ)^2.$ We prove a Cramér type moderate deviation expansion for $\mathbb{P}( \sum_{j=1}^k S_{j}^\circ /V_k^\circ \geq x)$ as $n\to \infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.

math.PR↗
arXiv2020

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

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.

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