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August 2016 Self-similar scaling limits of Markov chains on the positive integers
Jean Bertoin, Igor Kortchemski
Ann. Appl. Probab. 26(4): 2556-2595 (August 2016). DOI: 10.1214/15-AAP1157

Abstract

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If $X_{n}$ is such a Markov chain started at $n$, we establish a limit theorem for $\frac{1}{n}X_{n}$ appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process. We also study the asymptotic behavior of the time needed by $X_{n}$ to reach some fixed finite set. We identify three different regimes (roughly speaking the transient, the recurrent and the positive-recurrent regimes) in which $X_{n}$ exhibits different behavior. The present results extend those of Haas and Miermont [Bernoulli 17 (2011) 1217–1247] who focused on the case of nonincreasing Markov chains. We further present a number of applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks, nonnegative self-similar Markov processes, invariance principles for random walks conditioned to stay positive and exchangeable coalescence-fragmentation processes.

Citation

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Jean Bertoin. Igor Kortchemski. "Self-similar scaling limits of Markov chains on the positive integers." Ann. Appl. Probab. 26 (4) 2556 - 2595, August 2016. https://doi.org/10.1214/15-AAP1157

Information

Received: 1 December 2014; Revised: 1 September 2015; Published: August 2016
First available in Project Euclid: 1 September 2016

zbMATH: 1352.60103
MathSciNet: MR3543905
Digital Object Identifier: 10.1214/15-AAP1157

Subjects:
Primary: 60F17, 60G18, 60J10
Secondary: 60J35

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 4 • August 2016
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