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Journal Articles Journal of Theoretical Probability Year : 2018

Heavy-tailed random walks on complexes of half-lines

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 µ k. If χ k is 1 for one-sided half-lines k and 1/2 for two-sided half-lines, and α k is the tail exponent of the jumps on half-line k, we show that the recurrence classification for the case where all α k χ k ∈ (0, 1) is determined by the sign of ∑ k µ k cot(χ k πα k). 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 α 1 and α 2 ; our general setting exhibits the essential non-linearity 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).
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Dates and versions

hal-01376077 , version 1 (04-10-2016)

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Mikhail V Menshikov, Dimitri Petritis, Andrew R Wade. Heavy-tailed random walks on complexes of half-lines. Journal of Theoretical Probability, 2018, 31 (3), pp.1819-1859. ⟨10.1007/s10959-017-0753-5⟩. ⟨hal-01376077⟩
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