The elephant random walk (ERW) is a discrete-time random walk introduced by Schütz and Trimper (2004) in order to investigate how long-range memory affects the behavior of the random walk. Its particularity is that the next step of the walker depends on its whole past through a parameter p ∈ [ 0 , 1 ] . In this work, we investigate the validity of the central limit theorem of the ERW when the walker has only a gradually increasing memory. Our contribution provides a positive answer to a conjecture raised in a recent work by Gut and Stadtmüller (2022 Stat. Probab. Lett. 189 109598).

In this paper, we consider a two-dimension symmetric random walk with reset. We give, in the first part, some results about the distribution of every component. In the second part, we give some results about the final altitude $Z_n$. Finally, we analyse the statistical properties of $N_n^X$, the number of resets (the number of returns to state 1 after n steps) of the first component of the random walk. As a principal tool in these studies, we use the probability generating function.

The standard elephant random walk (ERW) is a discrete-time random walk introduced by Schütz and Trimper (2004 Phys. Rev. E 70 045101) with the particularity that the next step of the walker depends on its whole past through a parameter p ∈ [ 0 , 1 ] . In this note, we give a correction for the central limit theorem in the superdiffusive regime of the ERW with gradually increasing memory established by Aguech and El Machkouri (2024 J. Phys. A: Math. Theor. 57 065203, theorem 2). The mistake was pointed out by Roy et al (2024 arXiv:2403.02881, remark 1.1).

In this paper, we investigate an extended version of the elephant random walk model. Unlike the traditional approach where step sizes remain constant, our model introduces a novel feature: step sizes are generated as a sequence of positive independent and identically distributed random variables, and the step of the walker at time $ n+1 $ depends only on the steps of the walker between times $ 1, ..., m_n $, where $ (m_n)_{n\geqslant 1} $ is a sequence of positive integers growing to infinity as $ n $ goes to infinity. Our main results deal with the validity of the central limit theorem for this new variation of the standard ERW model introduced by Schütz and Trimper in $ 2004 $.