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Self-similar scaling limits of non-increasing Markov chains

Miermont, Grégory; Haas, Bénédicte (2011), Self-similar scaling limits of non-increasing Markov chains, Bernoulli, 17, 4, p. 1217-1247. http://dx.doi.org/10.3150/10-BEJ312

Type
Article accepté pour publication ou publié
External document link
http://arxiv.org/abs/0909.3764v1
Date
2011
Journal name
Bernoulli
Volume
17
Number
4
Publisher
Bernoulli Society
Pages
1217-1247
Publication identifier
http://dx.doi.org/10.3150/10-BEJ312
Metadata
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Author(s)
Miermont, Grégory
Haas, Bénédicte
Abstract (EN)
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 → ∞, 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 Λ-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 [1 1].
Subjects / Keywords
regular variation; self-similar Markov processes; regenerative compositions; Λ-coalescents; random walks with a barrier; absorption time

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