Remarks on the limiting behaviors of generalized elephant random walks

TL;DR

This paper investigates the long-term behavior of a generalized elephant random walk, analyzing how memory and drift parameters influence its scaling limits, including cases with degenerate parameters, and provides explicit examples.

Contribution

It introduces a generalized model incorporating memory and drift, analyzing its limiting behaviors and explicitly computing scaling factors in various scenarios.

Findings

Scaling factors depend on parameter behaviors

Degenerate parameters are accommodated in the analysis

Explicit examples of scaling factors are provided

Abstract

We study the limiting behaviors of a generalized elephant random walk on the integer lattice. This random walk is defined by using two sequences of parameters expressing the memory at each step from the whole past and the drift of each step to the right, respectively. This model is also regarded as a dependent Bernoulli process. Our results reveal how the scaling factors are determined by the behaviors of the parameters. In particular, we allow the degeneracy of the parameters. We further present several examples in which the scaling factors are explicitly computed.