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On the existence of quasi-stationary distributions in denumerable R-transient Markov chains

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Masaaki Kijima
Masaaki Kijima*
Affiliation:
The University of Tsukuba, Tokyo
*
∗ Postal address: Graduate School of Systems Management, The University of Tsukuba, Tokyo, 3–29–1 Otsuka, Bunkyo-ku, Tokyo 112, Japan.
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Abstract

Let {Xn, n = 0, 1, 2, ···} be a transient Markov chain which, when restricted to the state space 𝒩 + = {1, 2, ···}, is governed by an irreducible, aperiodic and strictly substochastic matrix 𝐏 = (pij), and let pij(n) = P ∈ Xn = j, Xk ∈ 𝒩+ for k = 0, 1, ···, n | X0 = i], i, j 𝒩 +. The prime concern of this paper is conditions for the existence of the limits, qij say, of as n →∞. If the distribution (qij) is called the quasi-stationary distribution of {Xn} and has considerable practical importance. It will be shown that, under some conditions, if a non-negative non-trivial vector x = (xi) satisfying rxT = xT𝐏 and exists, where r is the convergence norm of 𝐏, i.e. r = R–1 and and T denotes transpose, then it is unique, positive elementwise, and qij(n) necessarily converge to xj as n →∞. Unlike existing results in the literature, our results can be applied even to the R-null and R-transient cases. Finally, an application to a left-continuous random walk whose governing substochastic matrix is R-transient is discussed to demonstrate the usefulness of our results.

Keywords

SUBSTOCHASTIC MATRIXCONVERGENCE NORMLEFT INVARIANT VECTORCONDITIONAL TRANSITION PROBABILITYRATIO LIMIT THEOREMLEFT-CONTINUOUS RANDOM WALK

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 1 , March 1992 , pp. 21 - 36
Copyright
Copyright © Applied Probability Trust 1992 

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