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Conditioning an additive functional of a Markov chain to stay nonnegative. II. Hitting a high level

Part of: Probability theory on algebraic and topological structures Markov processes

Published online by Cambridge University Press:  01 July 2016

Saul D. Jacka ,
Zorana Lazic  and
Jon Warren
Saul D. Jacka*
Affiliation:
University of Warwick
Zorana Lazic*
Affiliation:
University of Warwick
Jon Warren*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
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Abstract

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Let (X t )t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v: E→ℝ\{0}, and let (φt )t≥0 be defined by φt =∫0 t v(X s )d s. We consider the case in which the process (φt )t≥0 is oscillating and that in which (φt )t≥0 has a negative drift. In each of these cases, we condition the process (X t ,φt )t≥0 on the event that (φt )t≥0 hits level y before hitting 0 and prove weak convergence of the conditioned process as y→∞. In addition, we show the relationship between the conditioning of the process (φt )t≥0 with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (φt )t≥0 with a negative drift to drift to ∞ and the conditioning of it to hit large levels before hitting 0.

Keywords

Markov chainconditional lawweak convergence

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60B10: Convergence of probability measures

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 1035 - 1055
Copyright
Copyright © Applied Probability Trust 2005 

References

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Jacka, S. D., Lazic, Z. and Warren, J. (2005). Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time. Adv. Appl. Prob. 37, 1015–1034.Google Scholar
Najdanovic, Z. (2003). Conditioning a Markov chain upon the behaviour of an additive functional. , University of Warwick.Google Scholar
Seneta, E. (1981). Nonnegative Matrices and Markov Chains, 2nd edn. Springer, New York.Google Scholar