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

Abstract

The one-dimensional elephant random walk is a typical model of discrete-time random walk with step-reinforcement, and is introduced by Schütz and Trimper (2004). It has a parameter α ∈ (−1, 1):The case α = 0 corresponds to the simple symmetric random walk, and when α > 0 (resp. α < 0), the mean displacement of the walker at time n grows (resp. vanishes) like n α . The walk admits a phase transition at α = 1/2 from the diffusive behavior to the superdiffusive behavior. In this paper, we study the rate of the moment convergence in the central limit theorem for the position of the walker when −1 < α ≤ 1/2. We find a crossover phenomenon in the rate of convergence of the 2m-th moments with m = 2, 3, . . . inside the diffusive regime −1 < α < 1/2.

Remarks on the limiting behaviors of generalized elephant random walks

“…As pointed in [13], a class of elephant random walks corresponds to the dependent Bernoulli sequences by simple relations. For the recent development on elephant random walks, see, e.g., [2,3,7,9].…”