Heavy-tailed random walks on complexes of half-lines

Menshikov, Mikhail V.; Petritis, Dimitri; Wade, Andrew R.

Abstract

We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution μk. If χk is 1 for one-sided half-lines k and 1 / 2 for two-sided half-lines, and αk is the tail exponent of the jumps on half-line k, we show that the recurrence classification for the case where all αkχk∈(0,1) is determined by the sign of ∑kμkcot(χkπαk). In the case of two half-lines, the model fits naturally on R and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in α1 and α2; our general setting exhibits the essential nonlinearity in the cotangent criterion. For the general model, we also show existence and non-existence of polynomial moments of return times. Our moments results are sharp (and new) for several cases of the oscillating random walk; they are apparently even new for the case of a homogeneous random walk on R with symmetric increments of tail exponent α∈(1,2).

Citation

Menshikov, M. V., Petritis, D., & Wade, A. R. (2018). Heavy-tailed random walks on complexes of half-lines. Journal of Theoretical Probability, 31(3), 1819-1859. https://doi.org/10.1007/s10959-017-0753-5

• Journal Article Type: Article
• Acceptance Date: Mar 6, 2017
• Online Publication Date: Mar 20, 2017
• Publication Date: Sep 1, 2018
• Deposit Date: Mar 13, 2017
• Publicly Available Date: Mar 13, 2017
• Journal: Journal of Theoretical Probability
• Print ISSN: 0894-9840
• Electronic ISSN: 1572-9230
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 31
• Issue: 3
• Pages: 1819-1859
• DOI: https://doi.org/10.1007/s10959-017-0753-5
• Public URL: https://durham-repository.worktribe.com/output/1391386