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.