The speed of a random walk excited by its recent history

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 leastM out of the N most recent jumps were to the right then the probability of jumping to the right is p1; however, if fewer thanM out of theN 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 )^l_i=1, above the pre-threshold level p0, as well as for one model with l = N such thresholds.