The speed of a random walk excited by its recent history. (March 2016)

• Record Type: Journal Article
• Authors: Pinsky, Ross G.
• Is Part Of: Advances in applied probability. Volume 48:Number 1(2016)
• Page Start: 215
• Page End: 234
• Publication Date: 2016-03
• Subjects: Random walk with internal states, -- excited random walk; Primary 60J10, -- 60F15; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2015.14
• Languages: English
• ISSNs: 0001-8678

Abstract

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.