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A ratio limit theorem for (sub) Markov chains on {1,2, …} with bounded jumps

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Harry Kesten
Harry Kesten*
Affiliation:
Cornell University
*
* Postal address: Department of Mathematics, White Hall, Cornell University, Ithaca, NY 14853–7901, USA.
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Abstract

We consider positive matrices Q, indexed by {1,2, …}. Assume that there exists a constant 1 L < ∞ and sequences u1< u2< · ·· and d1d2< · ·· such that Q(i, j) = 0 whenever i < ur < ur + L < j or i > dr + L > dr > j for some r. If Q satisfies some additional uniform irreducibility and aperiodicity assumptions, then for s > 0, Q has at most one positive s-harmonic function and at most one s-invariant measure µ. We use this result to show that if Q is also substochastic, then it has the strong ratio limit property, that is

for a suitable R and some R–1-harmonic function f and R–1-invariant measure µ. Under additional conditions µ can be taken as a probability measure on {1,2, …} and exists. An example shows that this limit may fail to exist if Q does not satisfy the restrictions imposed above, even though Q may have a minimal normalized quasi-stationary distribution (i.e. a probability measure µ for which R–1µ = µQ).

The results have an immediate interpretation for Markov chains on {0,1,2, …} with 0 as an absorbing state. They give ratio limit theorems for such a chain, conditioned on not yet being absorbed at 0 by time n.

Keywords

ABSORBING MARKOV CHAINUNIQUENESS OF HARMONIC FUNCTIONS AND HARMONIC MEASURESQUASI-STATIONARY DISTRIBUTION

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F99: None of the above, but in this section 60J25: Continuous-time Markov processes on general state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 3 , September 1995 , pp. 652 - 691
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Research supported by the NSF through a grant to Cornell University.

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