2013/05/30 by Ross G. Pinsky, Pinsky, Ross G.
Mathematics · #60F15 #60J10 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F15 #msc:60J10
paper · pdf · doi:10.48550/arxiv.1305.7242
This version contains a couple of additional results. Also, some small errors and imprecise statements have been corrected
arxiv created 2014/02/09 · arxiv updated 2014/02/11
Let N and M be positive integers satisfying 1≤ M≤ N, and let 0<p0<p1<1. Define a process \Xn\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 \frac MN→ 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, (Mi,pi)i=1l, above the pre-threshold level p0, as well as for one model with l=N such thresholds.