The Speed of a Random Walk Excited By Its Recent History

TL;DR

This paper analyzes the speed of a one-dimensional excited random walk influenced by recent history, deriving explicit formulas for its speed and examining the limiting behavior as the history window grows infinitely large.

Contribution

It introduces a model of a history-dependent random walk and provides explicit calculations for its speed and asymptotic behavior, extending previous models with multiple thresholds.

Findings

Explicit formula for the walk's speed.

Limiting speed as history window size approaches infinity.

Generalization to multiple thresholds and levels.

Abstract

Let $N$ and $M$ be positive integers satisfying $1\le M\le N$, and let $0<p_0<p_1<1$. Define a process $\{X_n\}_{n=0}^\infty$ on $\mathbb{Z}$ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first $N$ steps, the process behaves like a random walk that jumps to the right with probability $p_0$ and to the left with probability $1-p_0$. At subsequent steps the jump mechanism is defined as follows: if at least $M$ out of the $N$ most recent jumps were to the right, then the probability of jumping to the right is $p_1$; however, if fewer than $M$ out of the $N$ most recent jumps were to the right, then the probability of jumping to the right is $p_0$. We calculate the speed of the process. Then we let $N\to\infty$ and $\frac MN\to r\in[0,1]$, and calculate the limiting speed. More generally, we consider the above questions for a random walk with a finite number $l$ of threshold levels, $(M_i,p_i)_{i=1}^l$, above the pre-threshold level $p_0$, as well as for one model with $l=N$ such thresholds.