Related papers: A strong invariance principle for the elephant ran…
We study the so-called elephant random walk (ERW) which is a non-Markovian discrete-time random walk on $\mathbb{Z}$ with unbounded memory which exhibits a phase transition from diffusive to superdiffusive behaviour. We prove a law of large…
We introduce a new random walk with unbounded memory obtained as a mixture of the Elephant Random Walk and the Dynamic Random Walk which we call the Dynamic Elephant Random Walk (DERW). As a consequence of this mixture the distribution of…
The purpose of this paper is to investigate the asymptotic behavior of the multi-dimensional elephant random walk (MERW). It is a non-Markovian random walk which has a complete memory of its entire history. A wide range of literature is…
In this work, we discuss the smoothly amnesia-reinforced multidimensional elephant random walk (MARW). The scaling limit of the MARW is shown to exist in the diffusive, critical and superdiffusive regimes. We also establish the almost sure…
The randomized play-the-winner rule (RPW) is a response-adaptive design proposed by Wei and Durham (1978) for sequentially randomizing patients to treatments in a two-treatment clinical trial so that more patients are assigned to the better…
This paper is devoted to the asymptotic analysis of the reinforced elephant random walk (RERW) using a martingale approach. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and…
Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we…
The purpose of this paper is to establish, via a martingale approach, some refinements on the asymptotic behavior of the one-dimensional elephant random walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory parameter $p$…
In this article we shall derive functional limit theorems for the multi-dimensional elephant random walk (MERW) and thus extend the results provided for the one-dimensional marginal by Bercu and Laulin (2019). The MERW is a non-Markovian…
This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization.…
The elephant random walk (ERW) is a microscopic, one-dimensional, discrete-time, non-Markovian random walk, which can lead to anomalous diffusion due to memory effects. In this study, I propose a multi-dimensional generalization in which…
This paper investigates whether two independent Elephant Random Walks (ERWs) on $\mathbb{Z}$, each with a different memory parameter, can meet infinitely often, extending the work of Roy, Takei, and Tanemura. We also study the asymptotic…
In this paper, we introduce the elephant random walk (ERW) with memory consisting of randomly selected steps from its history. It is a time-changed variant of the standard elephant random walk with memory consisting of its full history. At…
The aim of this paper is to deepen the analysis of the asymptotic behavior of the so-called minimal random walk (MRW) using a new martingale approach. The MRW is a discrete-time random walk with infinite memory that has three regimes…
We explore the impact of long-range memory on the properties of a family of quantum walks in a one-dimensional lattice and discrete time, which can be understood as the quantum version of the classical "Elephant Random Walk" non-Markovian…
We consider a discrete-time random walk where the random increment at time step $t$ depends on the full history of the process. We calculate exactly the mean and variance of the position and discuss its dependence on the initial condition…
We study the enhanced diffusivity in the so called elephant random walk model with stops (ERWS) by including symmetric random walk steps at small probability $\epsilon$. At any $\epsilon > 0$, the large time behavior transitions from…
A step-reinforced random walk is a discrete-time non-Markovian process with long range memory. At each step, with a fixed probability p, the positively step-reinforced random walk repeats one of its preceding steps chosen uniformly at…
We consider a random walk on Z^d in an i.i.d. balanced random environment, that is a random walk for which the probability to jump from x to nearest neighbor x+e is the same as to nearest neighbor x-e. Assuming that the environment is…
We introduce an original way to estimate the memory parameter of the elephant random walk, a fascinating discrete time random walk on integers having a complete memory of its entire history. Our estimator is nothing more than a…