# Variations of the elephant random walk

## TL;DR

This paper investigates variations of the elephant random walk, focusing on models where the walker has limited memory, and extends the analysis to more general step sizes, providing new insights into memory-dependent random walks.

## Contribution

It introduces and analyzes new models of elephant random walks with restricted memory and generalized step sizes, extending previous work on memory-dependent stochastic processes.

## Key findings

- Characterization of walk behavior with limited memory
- Extension to generalized step size models
- Analogs of classical results for restricted-memory walks

## Abstract

In the classical simple random walk the steps are independent, viz., the walker has no memory. In contrast, in the elephant random walk which was introduced by Sch\"utz and Trimper in 2004, the walker remembers the whole past, and the next step always depends on the whole path so far. Our main aim is to prove analogous results when the elephant has only a restricted memory, for example remembering only the most remote step(s), the most recent step(s) or both. We also extend the models to cover more general step sizes.