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Computing absorbing times via fluid approximations

Part of: Extremal combinatorics Ergodic theory 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 T n 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 T n is well approximated by a deterministic time t n 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 t n that uses the spectral decomposition of the kernel of the chain as well as the asymptotic distribution of T n , 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

Information

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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