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Recurrence and Transience Criteria for Two Cases of Stable-Like Markov Chains

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  • Nikola Sandrić

    (University of Zagreb)

Abstract

We give recurrence and transience criteria for two cases of time-homogeneous Markov chains on the real line with transition kernel p(x,dy)=f x (y−x) dy, where f x (y) are probability densities of symmetric distributions and, for large |y|, have a power-law decay with exponent α(x)+1, with α(x)∈(0,2). If f x (y) is the density of a symmetric α-stable distribution for negative x and the density of a symmetric β-stable distribution for non-negative x, where α,β∈(0,2), then the chain is recurrent if and only if α+β≥2. If the function x↦f x is periodic and if the set {x:α(x)=α 0:=inf x∈ℝ α(x)} has positive Lebesgue measure, then, under a uniformity condition on the densities f x (y) and some mild technical conditions, the chain is recurrent if and only if α 0≥1.

Suggested Citation

  • Nikola Sandrić, 2014. "Recurrence and Transience Criteria for Two Cases of Stable-Like Markov Chains," Journal of Theoretical Probability, Springer, vol. 27(3), pages 754-788, September.
  • Handle: RePEc:spr:jotpro:v:27:y:2014:i:3:d:10.1007_s10959-012-0445-0
    DOI: 10.1007/s10959-012-0445-0
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    Cited by:

    1. Mikhail V. Menshikov & Dimitri Petritis & Andrew R. Wade, 2018. "Heavy-Tailed Random Walks on Complexes of Half-Lines," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1819-1859, September.
    2. Nikola Sandrić, 2016. "Ergodic Property of Stable-Like Markov Chains," Journal of Theoretical Probability, Springer, vol. 29(2), pages 459-490, June.

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