The speed of a general random walk reinforced by its recent history

TL;DR

This paper studies the behavior and limiting speed of various reinforced random walks whose step distributions depend on recent history, especially as the history window size grows large.

Contribution

It introduces and analyzes several variants of history-dependent random walks, focusing on their speed and asymptotic properties as the window size increases.

Findings

Derived formulas for the limiting speed as window size approaches infinity

Identified conditions under which the process exhibits certain speed behaviors

Provided insights into how recent history influences the walk's dynamics

Abstract

We consider several variants of a class of random walks whose increment distributions depend on the average value of the process over its most recent $N$ steps. We investigate the speed of the process, and in particular, the limiting speed as the "history window" $N\to\infty$.