# Heavy-Tailed Random Walks on Complexes of Half-Lines

Journal of Theoretical Probability, Mar 2017

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 $\mu _k$. If $\chi _k$ is 1 for one-sided half-lines k and 1 / 2 for two-sided half-lines, and $\alpha _k$ is the tail exponent of the jumps on half-line k, we show that the recurrence classification for the case where all $\alpha _k \chi _k \in (0,1)$ is determined by the sign of $\sum _k \mu _k \cot ( \chi _k \pi \alpha _k )$. In the case of two half-lines, the model fits naturally on ${{\mathbb {R}}}$ and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in $\alpha _1$ and $\alpha _2$; 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 ${{\mathbb {R}}}$ with symmetric increments of tail exponent $\alpha \in (1,2)$.

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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, 2017, 1-41, DOI: 10.1007/s10959-017-0753-5