Self-similar scaling limits of non-increasing Markov chains

Details Self-similar scaling limits of non-increasing Markov chains Self-similar scaling limits of non-increasing Markov chains

2009-09-21

arxiv.org/abs/0909.3764v2

Bernoulli 2011, Vol. 17, No. 4, 1217-1247

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/10-BEJ312 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ312

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 our paper, Scaling limits of Markov branching trees, with applications to Galton--Watson and random unordered trees (2010).

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