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Article Dans Une Revue Bernoulli Année : 2011

Self-similar scaling limits of non-increasing Markov chains

Résumé

We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from $n$ and appropriately rescaled, converges in distribution, as $n \rightarrow \infty$, to a non-increasing self-similar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the self-similar Markov process reaches first 0. We discuss various applications to the study of random walks with a barrier, of the number of collisions in $\Lambda$-coalescents that do not descend from infinity and of non-consistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in the forthcoming paper \cite{HMMarkovBranching}.
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Dates et versions

hal-00418741 , version 1 (21-09-2009)

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Bénédicte Haas, Grégory Miermont. Self-similar scaling limits of non-increasing Markov chains. Bernoulli, 2011, 17 (4), pp.1217-1247. ⟨hal-00418741⟩
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