Abstract
Let W be an integer-valued random variable satisfying \(E[W] =: \delta \ge 0\) and \(P(W<0)>0\), 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 \(x\in \mathbb Z\), 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 \(\delta \le 1\) and transient if \(\delta >1\). 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.
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Notes
The “cookie” terminology will be explained in Remark 1.1 below
Note that \(\Omega _{M,\mu }\) differs from \(\Omega _{M,\mu }^+\) in that we no longer require the first M cookies at each site to have nonnegative drift.
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J. Peterson was partially supported by NSA Grants H98230-13-1-0266 and H98230-15-1-0049.
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Davis, B., Peterson, J. Excited Random Walks with Non-nearest Neighbor Steps. J Theor Probab 30, 1255–1284 (2017). https://doi.org/10.1007/s10959-016-0697-1
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DOI: https://doi.org/10.1007/s10959-016-0697-1