arXiv subjects

Ahmad Darwiche

Publications and source records attributed to Ahmad Darwiche.

5 recordsLinked to original sources

arXiv2022

Compound Poisson approximation for simple transient random walks in random sceneries

Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(ξ(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(ξ(S_n))_{n\geq 0}$. Under suitable conditions, we make explicit the extremal index and show that the point process of exceedances converges to a compound Poisson point process. We give two examples for which the cluster size distribution can be made explicit.

math.PR↗

arXiv2022

Some properties on extremes for transient random walks in random sceneries

Let $(S_n)_{n \geq 0}$ be a transient random walk in the domain of attraction of a stable law and let $(ξ(s))_{s \in \mathbb{Z}}$ be a stationary sequence of random variables. In a previous work, under conditions of type $D(u_n)$ and $D'(u_n)$, we established a limit theorem for the maximum of the first $n$ terms of the sequence $(ξ(S_n))_{n\geq 0}$ as $n$ goes to infinity. In this paper we show that, under the same conditions and under a suitable scaling, the point process of exceedances converges to a Poisson point process. We also give some properties of $(ξ(S_n))_{n\geq 0}$.

math.PR↗

arXiv2022

Point processes of exceedances for random walks in random sceneries

Let $\{ξ(k), k \in \mathbb{Z} \}$ be a stationary sequence of random variables and let $\{S_n, n \in \mathbb{N}_+ \}$ be a transient random walk in the domain of attraction of a stable law. In the previous work \cite{Nicolas_Ahmad}, under conditions of type $D(u_n)$ and $D'(u_n)$ we provided a limit theorem for the maximum of the first $n$ terms of the sequence $\{ξ(S_n), n \in \mathbb{N} \}$. In this paper, under the same conditions we will see that, the limit of the process which counts the numbers of the exceedances of the form $\{ξ(S_k)>u_n\}, k\geq 1$, is a compound Poisson point process. We also deal with the so-called extremal index for the sequence $\{ξ(S_n), n \in \mathbb{N} \}$ and we discuss some weak mixing properties.

math.PR↗

arXiv2020

Convergence of weighted ergodic averages

Let $(X, \mathcal{A},μ)$ be a probability space and let $T$ be a contraction on $L^2(μ)$. We provide suitable conditions over sequences $(w_k)$, $(u_k)$ and $(A_k)$ in such a way that the weighted ergodic limit $\lim\limits_{N\rightarrow\infty}\frac{1}{A_N}\sum_{k=0}^{N-1} w_k T^{u_k}(f)=0$ $μ$-a.e. for any function $f$ in $L^2(μ)$. As a consequence of our main theorems, we also deal with the so-called one-sided weighted ergodic Hilbert transforms.

math.DS↗

arXiv2019

Extremes for transient random walks in random sceneries under weak independence conditions

Let $\{ξ(k), k \in \mathbb{Z} \}$ be a stationary sequence of random variables with conditions of type $D(u_n)$ and $D'(u_n)$. Let $\{S_n, n \in \mathbb{N} \}$ be a transient random walk in the domain of attraction of a stable law. We provide a limit theorem for the maximum of the first $n$ terms of the sequence $\{ξ(S_n), n \in \mathbb{N} \}$ as $n$ goes to infinity. This paper extends a result due to Franke and Saigo who dealt with the case where the sequence $\{ξ(k), k \in \mathbb{Z} \}$ is i.i.d.

math.PR↗