On Wasserstein-1 distance in the central limit theorem for elephant random walk

Abstract

Recently, the elephant random walk has attracted much attention. A wide range of literature studies are available for the asymptotic behavior of the process, such as central limit theorems, functional limit theorems, and the law of the iterated logarithm. However, there is no result concerning the Wasserstein-1 distance for normal approximations. In this paper, we show that the Wasserstein-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.

Rate of moment convergence in the central limit theorem for elephant random walks

“…There is a crossover phenomenon in the upper bounds at α = 0. A similar result on Wasserstein-1 distance is obtained by Ma, El Machkouri, and Fan [10]. Our main results give more precise information about such transition.…”

Gaussian fluctuations of the elephant random walk with gradually increasing memory

“…In the critical regime, we have (n log n) −1/2 S n → N (0, 1) in distribution and n 1−2p S n → L almost surely and in L 4 in the superdiffusive regime. The reader can refer to Baur and Bertoin [1], Bercu [2,3], Bercu and Laulin [4], Coletti et al [5], Laulin [11] and the references their in and to Ma et al [12] and Dedecker et al [6] for contributions to the rate of convergence in the central limit theorem of the ERW. Since the seminal paper by Schütz and Trimper [13], many variants of the ERW where introduced in the literature.…”

Rate of moment convergence in the central limit theorem for the elephant random walk

“…Their results suggest that there is a transition of the convergence rate at p = 1/2. However, the results obtained in [7,8] are about the change of upper bounds for the accuracy of the normal approximation. This motivates the study in the current paper.…”

“…The main interest of the study of the ERW is to investigate the long term memory effects. In [1], it is shown that the ERW exhibits both normal and anomalous (super) diffusion, depending on the memory parameter p. Last few years, many authors have studied several limit theorems describing how the memory influences the asymptotic behavior of the ERW (see [2][3][4][5][6][7][8][9] and references therein).…”

“…The central limit theorem for the diffusive regime p ∈ (0, 3/4) shows that the ERW is qualitatively similar to the ordinary random walk. Quite recently, more quantitative forms of the central limit theorem for p ∈ (0, 3/4] are obtained by [7,8]. Their results suggest that there is a transition of the convergence rate at p = 1/2.…”