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Xiequan Fan

Publications and source records attributed to Xiequan Fan.

At least 37 records · Page 2Linked to original sources
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.

math.PR↗
arXiv2020

Uniform Cramér moderate deviations and Berry-Esseen bounds 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 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 $n$.

math.PR↗
arXiv2019

Spatial-Frequency Domain Nonlocal Total Variation for Image Denoising

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.

eess.IV↗
arXiv2019

Large deviation inequalities for martingales in Banach spaces

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.

math.PR↗
arXiv2019

Cramér moderate deviation expansion for martingales with one-sided Sakhanenko's condition and its applications

We give a Cramér moderate deviation expansion for martingales with differences having finite conditional moments of order $2+ρ, ρ\in (0,1],$ and finite one-sided conditional exponential moments. The upper bound of the range of validity and the remainder of our expansion are both optimal. Consequently, it leads to a "half-side" moderate deviation principle for martingales. It is worth mentioning that our result is new even for independent random variables. Moreover, applications to quantile coupling inequality, $β$-mixing and $ψ$-mixing sequences are discussed.

math.PR↗
arXiv2019

Exponential inequalities for nonstationary Markov Chains

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.

stat.ML↗
arXiv2019

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

Let $(η_i)_{i\geq1}$ be a sequence of $ψ$-mixing random variables. Let $m=\lfloor n^α\rfloor, 0< α< 1, k=\lfloor n/(2m) \rfloor,$ and $Y_j = \sum_{i=1}^m η_{m(j-1)+i}, 1\leq j \leq k.$ Set $ S_k^o=\sum_{j=1}^{k } Y_j $ and $[S^o]_k=\sum_{i=1}^{k } (Y_j )^2.$ We prove a Cramér type moderate deviation expansion for $\mathbb{P}(S_k^o/\sqrt{[ S^o]_k} \geq x)$ as $n\to \infty.$ Our result is similar to the recent work of Chen\textit{ et al.}\ [Self-normalized Cramér-type moderate deviations under dependence. Ann.\ Statist.\ 2016; \textbf{44}(4): 1593--1617] where the authors established Cramér type moderate deviation expansions for $β$-mixing sequences. Comparing to the result of Chen \textit{et al.}, our results hold for mixing coefficients with polynomial decaying rate and wider ranges of validity.

math.PR↗
arXiv2018

Exact rates of convergence in some martingale central limit theorems

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 $2+ρ, ρ>0$. 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.

math.PR↗
arXiv2018

Tempered Fractional Multistable Motion and Tempered Multifractional Stable Motion

This work defines two classes of processes, that we term {\it tempered fractional multistable motion} and {\it 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.

math.PR↗
arXiv2018

Bernstein type inequalities for self-normalized martingales with applications

For self-normalized martingales with conditionally symmetric differences, de la Peña [A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27, No.1, 537-564] established the Gaussian type exponential inequalities. Bercu and Touati [Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 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 self-normalized sums, t-statistics and autoregressive processes are discussed.

math.PR↗
arXiv2018

Self-normalized Cramer type moderate deviations for martingales

Let $(ξ_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $S_n=\sum_{i=1}^nξ_i $ and $[ S]_n=\sum_{i=1}^n ξ_i^2.$ 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, 2003] for independent random variables.

math.PR↗
arXiv2017

Berry-Esseen bounds for self-normalized martingales

A Berry-Esseen bound is obtained for self-normalized martingales under the assumption of finite moments. The bound coincides with the classical Berry-Esseen bound for standardized martingales. An example is given to show the optimality of the bound. Applications to Student's statistic and autoregressive process are also discussed.

math.PR↗
arXiv2017

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

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.

math.PR↗
arXiv2016

Self-normalized deviation inequalities with application to $t$-statistics

Let $(ξ_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$ \mathbf{P} \Big( \max_{1\leq k \leq n} \sum_{i=1}^{k} |ξ_i|\big/ \big(\sum_{i=1}^{n} |ξ_i|^β\big)^{1/β} \geq x \Big) $$ for $x>0$ and $β>1.$ Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. An application to Student's $t$-statistics is also given.

math.PR↗
arXiv2016

On McDiarmid's inequality for Hamming distance

We improve the rate function of McDiarmid's inequality for Hamming distance. In particular, applying our result to the separately Lipschitz functions of independent random variables, we also refine the convergence rate function of McDiarmid's inequality around a median. Moreover, a non-uniform bound for the distance between the medians and the mean is also given. We also give some extensions of McDiarmid's inequalities to the case of nonnegative functionals of dependent random variables.

math.PR↗
arXiv2016

Nonuniform Berry-Esseen bounds for martingales with applications to statistical estimation

We establish nonuniform 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 $x\rightarrow \infty$. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.

math.PR↗