The speed of a random walk excited by its recent history

Abstract

Let N and M be positive integers satisfying 1≤ M≤ N, and let 0< p₀ < p₁ < 1. Define a process {X_n}_n=0^∞ on 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 p₀ and to the left with probability 1-p₀. 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 p₁; 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 p₀. We calculate the speed of the process. Then we let N→ ∞ and M/N→ r∈[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 p₀, as well as for one model with l=N such thresholds.