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Conditioning an additive functional of a markov chain to stay non-negative. I, Survival for a long time

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Jacka, Saul D., Lazic, Zorana and Warren, Jon (2005) Conditioning an additive functional of a markov chain to stay non-negative. I, Survival for a long time. Advances in Applied Probability, Vol.37 (No.4). pp. 1015-1034. ISSN 0001-8678

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Official URL: http://dx.doi.org/10.1239/aap/1134587751

Abstract

Let (X-t)(t >= 0) be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E -> R \ {0}, and let (phi(t))(t >= 0) be an additive functional defined by phi(t) = integral(0)(t)(X-s) ds. We consider the case in which the process (phi(t))(t >= 0) is oscillating and that in which (phi(t))(t >= 0) has a negative drift. In each of these cases, we condition the process (X-t, phi(t))(t >= 0) on the event that (phi(t))(t >= 0) is nonnegative until time T and prove weak convergence of the conditioned process as T -> infinity.

Item Type:

Submitted Journal Article

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science > Statistics

Library of Congress Subject Headings (LCSH):

Markov processes

Journal or Publication Title:

Advances in Applied Probability

Publisher:

Applied Probability Trust

ISSN:

0001-8678

Official Date:

December 2005

Dates:

Date	Event
December 2005	Published

Volume:

Vol.37

Number:

No.4

Number of Pages:

Page Range:

pp. 1015-1034

Identifier:

10.1239/aap/1134587751

Status:

Peer Reviewed

Publication Status:

Published

Access rights to Published version:

Restricted or Subscription Access

References:

[1] Barlow, M. T, Rogers, L. C. G., Williams, D. (1980). Wiener-Hopf factorization
for matrices. Seminaire de Probabilites XIV, Springer Lecture Notes in Math., 784,
324-331.
[2] Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative.
Ann. Probab. Vol. 22, No. 4, 2152-2167.
[3] Cohen, J. C. (1981). Convexity of the dominant eigenvalue of an essentially nonnegative
matrix. Proc. Amer. Math. Soc. Vol. 81, No. 4, 656-658.
[4] Doetsch, G. (1974). Introduction to the Theory and Application of the Laplace
Transformation. Berlin Heidelberg New York: Springer-Verlag.
[5] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. John
Wiley and Sons, Inc.
[6] Iglehart, D. L. (1974). Random walks with negative drift conditioned to stay positive.
J. Appl. Prob. 11, 742-751.
[7] Jacka, S. D., Lazic, Z., Warren, J. (2005) Conditioning an additive functional of a
Markov chain to stay non-negative II: hitting a high level. Submitted to J. Appl.
Prob.
[8] Jacka, S. D. and Roberts, G. O. (1988) Conditional diffusions: their infinitesimal
generators and limit laws. Research report, Dept. of Statistics, University of Warwick.
[9] Jacka, S. D. and Roberts, G. O. (1995) Weak convergence of conditioned processes
on a countable state space. J. Appl. Prob. 32, 902-916.
[10] Jacka, S. D. and Roberts, G. O. (1997) On strong forms of weak convergence. Stoch.
Proc. Their Apl. 67, 41-53.
[11] Jacka, S. D. and Warren, J. (2002) Examples of convergence and non-convergence
of Markov chains conditioned not to die. Electron. J. Probab. 7 (2002), No. 1, 22
p.
[12] Knight, F. B. (1969). Brownian local times and taboo processes. Trans. Amer.
Math. Soc. (143), 173–185.
[13] London, R. R., McKean, H. P., Rogers, L. C. G., Williams, D. (1980). A martingale
approach to some Wiener-Hopf problems II. Seminaire de Probabilites XVI,
Springer Lecture Notes in Math., 920, 68-91.
[14] Najdanovic, Z. (2003). Conditioning a Markov chain upon the behavoiur of an
additive functional. PhD Thesis, University of Warwick.
[15] Pinsky, R. G. (1985). On the convergence of diffusions processes conditioned to
remain in a bounded region for large time to limiting positive recurrent diffusion
processes. Ann. Probab. Vol. 13, No. 2, 363-378.
[16] Seneta, E. (1981). Non-negative Matrices and Markov Chains. New York Heidelberg
Berlin: Springer-Verlag.
[17] Wilkinson, J. H. (1965). The Algebraic Eigenvalue Problem. Clarendon Press Oxford.