Self-similar scaling limits of Markov chains on the positive integers

DOI10.1214/15-AAP1157zbMATH Open1352.60103arXiv1412.1068MaRDI QIDQ341630

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Publication date: 16 November 2016

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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 $frac1n{X}_n$ 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 & Miermont who focused on the case of non-increasing 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.

Full work available at URL: https://arxiv.org/abs/1412.1068

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