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Yaglom limits can depend on the starting state

Part of: Markov processes

Published online by Cambridge University Press:  20 March 2018

R. D. Foley  and
D. R. McDonald
R. D. Foley*
Affiliation:
Georgia Institute of Technology
D. R. McDonald*
Affiliation:
The University of Ottawa
*
* Postal address: Department of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA. Email address: [email protected]
** Postal address: Department of Mathematics and Statistics, The University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada. Email address: [email protected]
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Abstract

We construct a simple example, surely known to Harry Kesten, of an R-transient Markov chain on a countable state space S ∪ {δ}, where δ is absorbing. The transition matrix K on S is irreducible and strictly substochastic. We determine the Yaglom limit, that is, the limiting conditional behavior given nonabsorption. Each starting state x ∈ S results in a different Yaglom limit. Each Yaglom limit is an R-1-invariant quasi-stationary distribution, where R is the convergence parameter of K. Yaglom limits that depend on the starting state are related to a nontrivial R-1-Martin boundary.

Keywords

Quasi-stationaryR-transientρ-Martin boundaryYaglom limitgambler's ruineigenvalueeigenvectortime reversalDoob's h-transformationchange of measureinvariant measureharmonic functionsubstochastic

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J50: Boundary theory
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 1 - 34
Copyright
Copyright © Applied Probability Trust 2018 

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