The Speed of a Random Walk Excited By Its Recent History

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.