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The speed of a random walk excited by its recent history

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  24 March 2016

Ross G. Pinsky
Ross G. Pinsky*
Affiliation:
Technion – Israel Institute of Technology
*
* Postal address: Department of Mathematics, Technion – Israel Institute of Technology, Haifa, 32000, Israel. Email address: [email protected]
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Abstract

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 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, (Mi,pi) i=1l, above the pre-threshold level p0, as well as for one model with l=N such thresholds.

Keywords

Random walk with internal statesexcited random walk

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60F15: Strong theorems
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 215 - 234
Copyright
Copyright © Applied Probability Trust 2016 

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