The number of zeroes in Elephant random walks with delays

TL;DR

This paper studies how the number of delays, which can halt the process, evolves over time in Elephant Random Walks with memory, extending previous models to include delays and analyzing their impact.

Contribution

It introduces a model of Elephant Random Walks with delays and analyzes how delays accumulate over time, expanding understanding of memory-dependent stochastic processes.

Findings

Number of delays increases over time in ERWs with delays

Delays can significantly affect the walk's behavior

Extended previous models to include delays

Abstract

In the simple random walk the steps are independent, whereas in the Elephant Random Walk (ERW), which was introduced by Sch\"utz and Trimper in 2004, the next step always depends on the whole path so far. In an earlier paper we investigated Elephant Random Walks when the elephant has a restricted memory. Inspired by a suggestion by Bercu et al. (arXiv:1902.11220v1) we extended our results to the case when delays are allowed. In this paper we examine how the number of delays (that possibly stop the process) increases as time goes by.