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Self-similar scaling limits of non-increasing Markov chains (2009)
| Citations: | 11 - 1 self |
BibTeX
@MISC{Haas09self-similarscaling,
author = {Bénédicte Haas and Grégory Miermont},
title = {Self-similar scaling limits of non-increasing Markov chains },
year = {2009}
}
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Abstract
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 [11].
Keyphrases
non-increasing markov chain non-increasing self-similar markov process markov branching tree non-consistent regenerative composition non-negative integer random walk current state negative power self-similar markov process forthcoming paper various application large jump event rescaled absorption time
