arXiv subjects

Samir Ben Hariz

Publications and source records attributed to Samir Ben Hariz.

4 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↗

arXiv2026

Quantitative central limit theorem for an integrated periodogram via the fourth moment theorem

We revisit the central limit theorem for integrated periodograms, equivalently for Toeplitz quadratic forms of stationary Gaussian sequences. Under a regular-variation assumption allowing long-memory singularities and slowly varying corrections, we prove a quantitative central limit theorem in 1-Wasserstein distance. The proof uses a second Wiener chaos representation and the Malliavin-Stein method (in particular, the Fourth Moment Theorem), reducing normal approximation to (i) variance asymptotics and (ii) an explicit control of the fourth cumulant via trace estimates for an associated integral operator. For convenience, we provide self-contained kernel estimates (Dirichlet-type bounds, convolution inequalities, and a weighted Schur test) used in the argument.

math.PR↗

arXiv2022

Fast calibration of weak FARIMA models

In this paper, we investigate the asymptotic properties of Le Cam's one-step estimator for weak Fractionally AutoRegressive Integrated Moving-Average (FARIMA) models. For these models, noises are uncorrelated but neither necessarily independent nor martingale differences errors. We show under some regularity assumptions that the one-step estimator is strongly consistent and asymptotically normal with the same asymptotic variance as the least squares estimator. We show through simulations that the proposed estimator reduces computational time compared with the least squares estimator. An application for providing remotely computed indicators for time series is proposed.

math.ST↗

arXiv2007

Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences

Let $(X_i)_{i=1,...,n}$ be a possibly nonstationary sequence such that $\mathscr{L}(X_i)=P_n$ if $i\leq nθ$ and $\mathscr{L}(X_i)=Q_n$ if $i>nθ$, where $0<θ<1$ is the location of the change-point to be estimated. We construct a class of estimators based on the empirical measures and a seminorm on the space of measures defined through a family of functions $\mathcal{F}$. We prove the consistency of the estimator and give rates of convergence under very general conditions. In particular, the $1/n$ rate is achieved for a wide class of processes including long-range dependent sequences and even nonstationary ones. The approach unifies, generalizes and improves on the existing results for both parametric and nonparametric change-point estimation, applied to independent, short-range dependent and as well long-range dependent sequences.

math.ST↗