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Wald's equations, first passage times and moments of ladder variables in Markov random walks

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Cheng Der Fuh  and
Tze Leung Lai
Cheng Der Fuh*
Affiliation:
Academia Sinica
Tze Leung Lai*
Affiliation:
Stanford University
*
∗Postal address: Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, ROC. Email address: [email protected].
∗∗Postal address: Department of Statistics, Stanford University, Stanford, CA 94305, USA.
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Abstract

Previous work in extending Wald's equations to Markov random walks involves finiteness of moment generating functions and uniform recurrence assumptions. By using a new approach, we can remove these assumptions. The results are applied to establish finiteness of moments of ladder variables and to derive asymptotic expansions for expected first passage times of Markov random walks. Wiener–Hopf factorizations for Markov random walks are also applied to analyse ladder variables and related first passage problems.

Keywords

First passage timeladder height distributionWiener-Hopf factorizationMarkov random walkWald's equations

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 566 - 580
Copyright
Copyright © Applied Probability Trust 1998 

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References

Alsmeyer, G. (1994). On the Markov renewal theorem. Stoch. Proc. Appl. 50, 37–56.CrossRefGoogle Scholar
Arjas, E., and Speed, T. P. (1973). Symmetric Wiener–Hopf factorizations in Markov additive processes. Z. Wahrscheinlichkeitsth. 26, 105–118.Google Scholar
Asmussen, S. (1989). Aspects of matrix Wiener–Hopf factorization in applied probability. Math. Scientist 14, 101–116.Google Scholar
Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York.Google Scholar
Chow, Y. S., and Lai, T. L. (1979). Moments of ladder variables for driftless random walks. Z. Wahrscheinlichkeitsth. 48, 253–257.Google Scholar
Chow, Y. S., and Teicher, H. (1988). Probability Theory, 2nd edn. Springer, New York.Google Scholar
Fuh, C. D. (1997). Corrected diffusion approximations for ruin probabilities in Markov random walk. Adv. Appl. Prob. 29, 695–712.Google Scholar
Gut, A. (1974). On the moments and limit distributions of some first passage times. Ann. Prob. 2, 277–308.Google Scholar
Gut, A. (1988). Stopped Random Walks. Springer, New York.CrossRefGoogle Scholar
Jensen, J. L. (1987). A note on asymptotic expansions for Markov chains using operator theory. Adv. Appl. Math. 8, 377–392.Google Scholar
Jensen, J. L. (1989). Asymptotic expansions for strongly mixing Harris recurrent Markov chains. Scand. J. Statist. 16, 47–63.Google Scholar
Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Prob. 2, 355–386.Google Scholar
Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.CrossRefGoogle Scholar
Miller, H. D. (1962). Absorption probabilities for sums of random variables defined on a finite Markov chain. Proc. Camb. Phil. Soc. 58, 286–298.Google Scholar
Nagaev, S. V. (1957). Some limit theorems for stationary Markov chains. Theory Prob. Appl. 2, 378–406.Google Scholar
Pacheco, A., and Prabhu, N. U. (1995). Markov-additive processes of arrivals. In Advances in Queueing, CRC Press, Boca Raton, pp. 167–194.Google Scholar
Pitman, J. W. (1974). An identity for stopping times of a Markov process. In E. J. Williams: Studies in Probability and Statistics, Jerusalem Academic Press, pp. 41–57.Google Scholar
Prabhu, N. U., Tang, L. C., and Zhu, Y. (1991). Some new results for the Markov random walk. J. Math. Phys. Sci. 25, 635–663.Google Scholar
Sadowsky, J. S. (1989). A dependent data extension of Wald's identity and its applications to sequential test performance computation. IEEE Trans. Inform. Theory 35, 834–842.Google Scholar
Siegmund, D. (1985). Sequential Analysis. Springer, New York.Google Scholar