On the limit law of the superdiffusive elephant random walk

TL;DR

This paper investigates the asymptotic properties of the superdiffusive elephant random walk's limit distribution, revealing tail asymmetry, asymptotic moments, and special function representations.

Contribution

It derives explicit asymptotic expansions and tail behaviors of the limit distribution using special functions, advancing understanding of superdiffusive random walks.

Findings

Limit variable has a density with asymmetric tails.

Asymptotic moments are derived using recurrence relations.

The distribution exhibits unimodality and log-concavity under certain conditions.

Abstract

When the memory parameter of the elephant random walk is above a critical threshold, the process becomes superdiffusive and, once suitably normalised, converges to a non-Gaussian random variable. In a recent paper by the three first authors, it was shown that this limit variable has a density and that the associated moments satisfy a nonlinear recurrence relation. In this work, we exploit this recurrence to derive an asymptotic expansion of the moments and the asymptotic behaviour of the density at infinity. In particular, we show that an asymmetry in the distribution of the first step of the random walk leads to an asymmetry of the tails of the limit variable. These results follow from a new, explicit expression of the Stieltjes transformation of the moments in terms of special functions such as hypergeometric series and incomplete beta integrals. We also obtain other results about the random variable, such as unimodality and, for certain values of the memory parameter, log-concavity.