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Markov chains conditioned never to wait too long at the origin

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Jacka, Saul D.. (2009) Markov chains conditioned never to wait too long at the origin. Journal of Applied Probability, Vol.46 (No.3). pp. 812-826. ISSN 0021-9002

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Official URL: http://dx.doi.org/10.1239/jap/1253279853

Abstract

Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by τ the first time that the chain, X, waits for at least one unit of time at the origin, we consider conditioning the chain on the event (τ›T). We show that there is a weak limit as T→∞ in the cases where either the state space is finite or X is transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than τ and τ is subexponential.

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science > Statistics

Library of Congress Subject Headings (LCSH):

Markov processes, Probabilities, Branching processes, Boundary value problems, Convergence

Journal or Publication Title:

Journal of Applied Probability

Publisher:

Applied Probability Trust

ISSN:

0021-9002

Official Date:

September 2009

Dates:

Date	Event
September 2009	Published

Volume:

Vol.46

Number:

No.3

Page Range:

pp. 812-826

Identification Number:

10.1239/jap/1253279853

Status:

Peer Reviewed

Access rights to Published version:

Open Access

References:

Doney, R. A. and Bertoin, J. (1996). Some asymptotic results for transient random walks. Adv. App. Prob. 28, 207--226.
Mathematical Reviews (MathSciNet): MR1372336
Zentralblatt MATH: 0854.60069
Digital Object Identifier: doi:10.2307/1427918
JSTOR: links.jstor.org
Doney, R. A. and Bertoin, J. (1997). Spitzer's condition for random walks and Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 33, 167--178.
Mathematical Reviews (MathSciNet): MR1443955
Digital Object Identifier: doi:10.1016/S0246-0203(97)80120-3
Doney, R. A. and Chaumont, L. (2005). On Lévy processes conditioned to stay positive. Electron. J. Prob. 10, 948--961.
Mathematical Reviews (MathSciNet): MR2164035
Feller, W. (1968). An Introduction To Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York.
Mathematical Reviews (MathSciNet): MR228020
Feller, W. (1971). An Introduction To Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
Mathematical Reviews (MathSciNet): MR270403
Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Prob. 23, 501--521.
Mathematical Reviews (MathSciNet): MR1334159
Zentralblatt MATH: 0827.60061
Digital Object Identifier: doi:10.1214/aop/1176988277
Project Euclid: euclid.aop/1176988277
Jacka, S. D. and Roberts, G. O. (1994). Strong forms of weak convergence. Stoch. Process. Appl. 67, 41--53.
Mathematical Reviews (MathSciNet): MR1445043
Zentralblatt MATH: 0890.60005
Digital Object Identifier: doi:10.1016/S0304-4149(97)00002-1
Jacka, S. D. and Roberts, G. O. (1995). Weak convergence of conditioned processes on a countable state space. J. Appl. Prob. 32, 902--916.
Mathematical Reviews (MathSciNet): MR1363332
Zentralblatt MATH: 0839.60069
Digital Object Identifier: doi:10.2307/3215203
JSTOR: links.jstor.org
Jacka, S. and Warren, J. (2002). Examples of convergence and non-convergence of Markov chains conditioned not to die. Electron. J. Prob. 7, 22pp.
Mathematical Reviews (MathSciNet): MR1887621
Zentralblatt MATH: 1014.60074
Jacka, S., 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.
Mathematical Reviews (MathSciNet): MR2193994
Zentralblatt MATH: 1101.60056
Digital Object Identifier: doi:10.1239/aap/1134587751
Project Euclid: euclid.aap/1134587751
Jacka, S., Lazic, Z. and Warren, J. (2005). Conditioning an additive functional of a Markov chain to stay nonnegative. II. Hitting a high level. Adv. Appl. Prob. 37, 1035--1055.
Mathematical Reviews (MathSciNet): MR2193995
Zentralblatt MATH: 1152.60340
Digital Object Identifier: doi:10.1239/aap/1134587752
Project Euclid: euclid.aap/1134587752
Kesten, H. (1995). A ratio limit theorem for (sub) Markov chains on $\1,2,\ldots\$ with bounded jumps. Adv. Appl. Prob. 27, 652--691.
Mathematical Reviews (MathSciNet): MR1341881
Zentralblatt MATH: 0829.60059
Digital Object Identifier: doi:10.2307/1428129
JSTOR: links.jstor.org
Kyprianou, A. E. and Palmowski, Z. (2006). Quasi-stationary distributions for Lévy processes. Bernoulli 12, 571--581.
Mathematical Reviews (MathSciNet): MR2248228
Digital Object Identifier: doi:10.3150/bj/1155735927
Project Euclid: euclid.bj/1155735927
Roberts, G. O. and Jacka, S. D. (1994). Weak convergence of conditioned birth and death processes. J. Appl. Prob. 31, 90--100.
Mathematical Reviews (MathSciNet): MR1260573
Zentralblatt MATH: 0796.60077
Digital Object Identifier: doi:10.2307/3215237
JSTOR: links.jstor.org
Roberts, G. O., Jacka, S. D. and Pollett, P. K. (1997). Non-explosivity of limits of conditioned birth and death processes. J. Appl. Prob. 34, 35--45.
Mathematical Reviews (MathSciNet): MR1429052
Zentralblatt MATH: 0874.60070
Digital Object Identifier: doi:10.2307/3215172
JSTOR: links.jstor.org
Seneta, E. (1981). Nonnegative Matrices and Markov Chains. Springer, New York.
Mathematical Reviews (MathSciNet): MR719544
Seneta, E. and Vere-Jones, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403--434.
Mathematical Reviews (MathSciNet): MR207047
Zentralblatt MATH: 0147.36603
Digital Object Identifier: doi:10.2307/3212128
JSTOR: links.jstor.org
Sigman, K. (1999). Appendix: a primer on heavy-tailed distributions. Queues with heavy-tailed distributions. Queueing Systems 33, 261--275.
Mathematical Reviews (MathSciNet): MR1748646
Digital Object Identifier: doi:10.1023/A:1019180230133
Williams, D. (1979). Diffusions, Markov Processes, and Martingales, Vol. I. John Wiley, Chichester.
Mathematical Reviews (MathSciNet): MR531031