Excited Random Walks with Non-nearest Neighbor Steps

Burgess Davis () and Jonathon Peterson ()
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Burgess Davis: Purdue University
Jonathon Peterson: Purdue University

Journal of Theoretical Probability, 2017, vol. 30, issue 4, 1255-1284

Abstract: Abstract Let W be an integer-valued random variable satisfying $$E[W] =: \delta \ge 0$$ E [ W ] = : δ ≥ 0 and $$P(W 0$$ P ( W 0 , and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any integer $$x\in \mathbb Z$$ x ∈ Z , the size of the next step is an independent random variable with the same distribution as W. We show that this self-interacting random walk is recurrent if $$\delta \le 1$$ δ ≤ 1 and transient if $$\delta >1$$ δ > 1 . This is a special case of our main result which concerns the recurrence and transience of excited random walks (or cookie random walks) with non-nearest neighbor jumps.

Keywords: Excited random walk; Cookie random walk; Recurrence; Transience; Primary 60K35; Secondary 60K37; 60J15 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-016-0697-1

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