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.
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Acknowledgements
The author would like to thank Prof. Zoran Vondraček for many discussions on the topic and for helpful comments on the presentation of the results. The author also thanks the referees for careful reading and helpful comments.
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Sandrić, N. Recurrence and Transience Criteria for Two Cases of Stable-Like Markov Chains. J Theor Probab 27, 754–788 (2014). https://doi.org/10.1007/s10959-012-0445-0
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DOI: https://doi.org/10.1007/s10959-012-0445-0
Keywords
- Characteristics of semimartingale
- Feller process
- Harris recurrence
- Markov chain
- Markov process
- Recurrence
- Stable distribution
- Stable-like process
- T-model
- Transience