The Annals of Applied Probability

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

Jean Bertoin and Igor Kortchemski

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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.

Article information

Ann. Appl. Probab., Volume 26, Number 4 (2016), 2556-2595.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G18: Self-similar processes
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Markov chains self-similar Markov processes Lévy processes invariance principles


Bertoin, Jean; Kortchemski, Igor. Self-similar scaling limits of Markov chains on the positive integers. Ann. Appl. Probab. 26 (2016), no. 4, 2556--2595. doi:10.1214/15-AAP1157.

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