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March 2016 The speed of a random walk excited by its recent history
Ross G. Pinsky
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Adv. in Appl. Probab. 48(1): 215-234 (March 2016).

Abstract

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.

Citation

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Ross G. Pinsky. "The speed of a random walk excited by its recent history." Adv. in Appl. Probab. 48 (1) 215 - 234, March 2016.

Information

Published: March 2016
First available in Project Euclid: 8 March 2016

zbMATH: 1337.60085
MathSciNet: MR3473575

Subjects:
Primary: 60F15, 60J10

Rights: Copyright © 2016 Applied Probability Trust

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Vol.48 • No. 1 • March 2016
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