Advances in Applied Probability

The speed of a random walk excited by its recent history

Ross G. Pinsky

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Let N and M be positive integers satisfying 1≤ MN, 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 M/Nr∈[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.

Article information

Adv. in Appl. Probab., Volume 48, Number 1 (2016), 215-234.

First available in Project Euclid: 8 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60F15: Strong theorems

Random walk with internal states excited random walk


Pinsky, Ross G. The speed of a random walk excited by its recent history. Adv. in Appl. Probab. 48 (2016), no. 1, 215--234.

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