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Computing absorbing times via fluid approximations

Part of: Ergodic theory Extremal combinatorics Markov processes

Published online by Cambridge University Press:  08 September 2017

Nicolas Gast  and
Bruno Gaujal
Nicolas Gast*
Affiliation:
Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LIG, 38000 Grenoble, France
Bruno Gaujal*
Affiliation:
Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LIG, 38000 Grenoble, France
*
* Postal address: Institute of Engineering, Université Grenoble Alpes, Inria, Bâtiment IMAG, 700 avenue Centrale, 38400 St Martin d'Heres, France.
* Postal address: Institute of Engineering, Université Grenoble Alpes, Inria, Bâtiment IMAG, 700 avenue Centrale, 38400 St Martin d'Heres, France.
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Abstract

In this paper we compute the absorbing time Tn of an n-dimensional discrete-time Markov chain comprising n components, each with an absorbing state and evolving in mutual exclusion. We show that the random absorbing time Tn is well approximated by a deterministic time tn that is the first time when a fluid approximation of the chain approaches the absorbing state at a distance 1 / n. We provide an asymptotic expansion of tn that uses the spectral decomposition of the kernel of the chain as well as the asymptotic distribution of Tn, relying on extreme values theory. We show the applicability of this approach with three different problems: the coupon collector, the erasure channel lifetime, and the coupling times of random walks in high-dimensional spaces.

Keywords

Markov chainfluid approximationextreme valuecoupon collectorcoupling time

MSC classification

Primary: 05D40: Probabilistic methods
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 37A25: Ergodicity, mixing, rates of mixing
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 768 - 790
Copyright
Copyright © Applied Probability Trust 2017 

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