Self-similar scaling limits of non-increasing Markov chains

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 $n\to \infty$, 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).

Published in at http://dx.doi.org/10.3150/10-BEJ312 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (4)

• Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees
• Asymptotic laws for compositions derived from transformed subordinators
• Spinal partitions and invariance under re-rooting of continuum random trees
• On the number of collisions in beta(2, $b$)-coalescents

Cited by in corpus (7)

• Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees
• Random walks with preferential relocations and fading memory: a study through random recursive trees
• The Mittag-Leffler process and a scaling limit for the block counting process of the Bolthausen-Sznitman coalescent
• Precise asymptotics for the density and the upper tail of exponential functionals of subordinators
• On the asymptotics of moments of linear random recurrences
• Lambda-coalescents with dust component
• Regenerative tree growth: structural results and convergence