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Mohamed El Machkouri

Publications and source records attributed to Mohamed El Machkouri.

15 recordsLinked to original sources
arXiv2026

On a class of unbalanced step-reinforced random walks

A step-reinforced random walk is a discrete-time stochastic process with long-range dependence. At each step, with a fixed probability $α$, the so-called positively step-reinforced random walk repeats one of its previous steps, chosen randomly and uniformly from its entire history. Alternatively, with probability $1-α$, it makes an independent move. For the so-called negatively step-reinforced random walk, the process is similar, but any repeated step is taken with its direction reversed. These random walks have been introduced respectively by Simon (1955) and Bertoin (2024) and are sometimes refered to the self-confident step-reinforced random walk and the counterbalanced step-reinforced random walk respectively. In this work, we introduce a new class of unbalanced step-reinforced random walks for which we prove the strong law of large numbers and the central limit theorem. In particular, our work provides a unified treatment of the elephant random walk introduced by Schutz and Trimper (2004) and the positively and negatively step-reinforced random walks.

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↗
arXiv2018

Stable limits for Markov chains via the Principle of Conditioning

We study limit theorems for partial sums of instantaneous functions of a homogeneous Markov chain on a general state space. The summands are heavy-tailed and the limits are stable distributions. The conditions imposed on the transition operator $P$ of the Markov chain ensure that the limit is the same as if the summands were independent. Such a~scheme admits a physical interpretation, as given in Jara et al. (Ann. Appl. Probab., 19 (2009), 2270--2300). We considerably extend the results of Jara et al., (ibid.) and Cattiaux and Manou-Abi (ESAIM Probab. Stat., 18 (2014), 468--486). We show that the theory holds under the assumption of operator uniform integrability in $L^2$ of $P$ (a notion introduced by Wu (J. Funct. Anal., 172 (2000), 301--376)) plus the $L^2$-spectral gap property. If we strengthen the uniform integrability in $L^2$ to the hyperboundedness, then the $L^2$-spectral gap property can be relaxed to the strong mixing at geometric rate (in practice: to geometric ergodicity). We provide an example of a Markov chain on a countable space that is uniformly integrable in $L^2$ (and admits an $L^2$-spectral gap), while it is not hyperbounded. Moreover, we show by example that hyperboundedness is still a weaker property than $ϕ$-mixing, what enlarges the range of models of interest. What makes our assumptions working is a new, efficient version of the Principle of Conditioning that operates with conditional characteristic functions rather than predictable characteristics.

math.PR↗
arXiv2016

Least squares estimator for non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes

The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein-Uhlenbeck process defined as $dX_t=θX_tdt+dG_t,\ t\geq0$ with an unknown parameter $θ>0$, where $G$ is a Gaussian process. We provide sufficient conditions, based on the properties of $G$, ensuring the strong consistency and the asymptotic distribution of our estimator $\widetildeθ_t$ of $θ$ based on the observation $\{X_s,\ s\in[0,t]\}$ as $t\rightarrow\infty$. Our approach offers an elementary, unifying proof of \cite{BEO}, and it allows to extend the result of \cite{BEO} to the case when $G$ is a fractional Brownian motion with Hurst parameter $H\in(0,1)$. We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.

math.PR↗
arXiv2014

Orthomartingale-coboundary decomposition for stationary random fields

We provide a new projective condition for a stationary real random field indexed by the lattice $\Z^d$ to be well approximated by an orthomartingale in the sense of Cairoli (1969). Ourmain result can be viewed as a multidimensional version of the martingale-coboundary decomposition method which the idea goes back to Gordin (1969). It is a powerfull tool for proving limit theorems or large deviations inequalities for stationary random fields when the corresponding result is valid for orthomartingales.

math.PR↗
arXiv2014

Kernel density estimation for stationary random fields

In this paper, under natural and easily verifiable conditions, we prove the $\mathbb{L}^1$-convergence and the asymptotic normality of the Parzen-Rosenblatt density estimator for stationary random fields of the form $X_k = g\left(\varepsilon_{k-s}, s \in \Z^d \right)$, $k\in\Z^d$, where $(\varepsilon_i)_{i\in\Z^d}$ are i.i.d real random variables and $g$ is a measurable function defined on $\R^{\Z^d}$. Such kind of processes provides a general framework for stationary ergodic random fields. A Berry-Esseen's type central limit theorem is also given for the considered estimator.

math.ST↗
arXiv2013

On the asymptotic normality of frequency polygons for strongly mixing spatial processes

This paper establishes the asymptotic normality of frequency polygons in the context of stationary strongly mixing random fields indexed by $\Z^d$. Our method allows us to consider only minimal conditions on the width bins and provides a simple criterion on the mixing coefficients. In particular, we improve in several directions a previous result by Carbon, Francq and Tran (2010).

math.ST↗
arXiv2012

A central limit theorem for stationary random fields

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form $X_k = g(\varepsilon_{k-s}, s \in \Z^d)$, $k\in\Z^d$, where $(\varepsilon_i)_{i\in\Z^d}$ are i.i.d random variables and $g$ is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying domain on which the process is observed. A limit theorem for the sample auto-covariance function is also established.

math.PR↗
arXiv2012

Kernel deconvolution estimation for random fields

In this work, we establish the asymptotic normality of the deconvolution kernel density estimator in the context of strongly mixing random fields. Only minimal conditions on the bandwidth parameter are required and a simple criterion on the strong mixing coefficients is provided. Our approach is based on the Lindeberg's method rather than on Bernstein's technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. We deal also with nonmixing random fields which can be written as a (nonlinear) functional of i.i.d. random fields by considering the physical dependence measure coefficients introduced by Wu (2005).

math.ST↗
arXiv2010

Asymptotic normality of the Parzen-Rosenblatt density estimator for strongly mixing random fields

We prove the asymptotic normality of the kernel density estimator (introduced by Rosenblatt (1956) and Parzen (1962)) in the context of stationary strongly mixing random fields. Our approach is based on the Lindeberg's method rather than on Bernstein's small-block-large-block technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. Our method allows us to consider only minimal conditions on the bandwidth parameter and provides a simple criterion on the (non-uniform) strong mixing coefficients which do not depend on the bandwith.

math.ST↗
arXiv2009

Asymptotic normality of kernel estimates in a regression model for random fields

We establish the asymptotic normality of the regression estimator in a fixed-design setting when the errors are given by a field of dependent random variables. The result applies to martingale-difference or strongly mixing random fields. On this basis, a statistical test that can be applied to image analysis is also presented.

math.ST↗
arXiv2005

Nonparametric regression estimation for random fields in a fixed-design

We investigate the nonparametric estimation for regression in a fixed-design setting when the errors are given by a field of dependent random variables. Sufficient conditions for kernel estimators to converge uniformly are obtained. These estimators can attain the optimal rates of uniform convergence and the results apply to a large class of random fields which contains martingale-difference random fields and mixing random fields.

math.ST↗
arXiv2004

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

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 $n^{-1/2}\log n$ is reached. We give interesting examples of martingales with unbounded increments which belong to the considered class.

math.PR↗
arXiv2004

On the central and local limit theorem for martingale difference sequences

Let $(Ω, \A, μ)$ be a Lebesgue space and $T$ an ergodic measure preserving automorphism on $Ω$ with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on $Ω$ with a common non-degenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.

math.PR↗