The speed of a random walk excited by its recent history. (March 2016) - Pinsky, Ross G.
Abstract: Let N and M be positive integers satisfying 1≤ M ≤ N, and let 0< p 0 < p 1 < 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 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 → ∞ 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 p 0, as well as for one model with l = N such thresholds.