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Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Cheng-Der Fuh
Cheng-Der Fuh*
Affiliation:
National Central University and Academia Sinica
*
∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, ROC. Email address: stcheng@stat.sinica.edu.tw
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Abstract

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Let {(Xn, Sn), n ≥ 0} be a Markov random walk in which Xn takes values in a general state space and Sn takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

Keywords

Boundary crossing probabilityladder height distributionMarkov-dependent Wald martingaleovershootPoisson equationuniform Markov renewal theory

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K15: Markov renewal processes, semi-Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 3 , September 2007 , pp. 826 - 852
Copyright
Copyright © Applied Probability Trust 2007 

References

Alsmeyer, G. (1997). The Markov renewal theorem and related results. Markov Proc. Relat. Fields 3, 103–127.Google Scholar
Alsmeyer, G. (2000). The ladder variables of a Markov random walk. Theoret. Prob. Math. Statist. 20, 151–168.Google Scholar
Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley, Reading, MA.Google Scholar
Arndt, K. (1980). Asymptotic properties of the distribution of the supremum of a random walk on a Markov chain. Theory Prob. Appl. 25, 309–323.Google Scholar
Asmussen, S. (1989a). Aspects of matrix Wiener–Hopf factorization in applied probability. Math. Scientist. 14, 101–116.Google Scholar
Asmussen, S. (1989b). Risk theory in a Markov environment. Scand. Actuarial J.. 1989, 69–100.Google Scholar
Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues. Springer, New York.Google Scholar
Basseville, M. and Nikiforov, I. V. (1993). Detection of Abrupt Changes: Theory and Application. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Blanchet, J. and Glynn, P. (2006). Complete corrected diffusion approximations for the maximum of a random walk. Ann. Appl. Prob. 16, 951–983.Google Scholar
Burman, D. Y. and Smith, D. R. (1986). An asymptotic analysis of a queueing system with Markov-modulated arrivals. Operat. Res. 34, 105–119.CrossRefGoogle Scholar
Chan, H. P. and Lai, T. L. (2003). Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks. Ann. Appl. Prob. 13, 395–429.Google Scholar
Chang, J. T. (1992). On moments of the first ladder height of random walks with small drift. Ann. Appl. Prob. 2, 714–738.Google Scholar
Chang, J. T. and Peres, Y. (1997). Ladder heights, Gaussian random walks and the Riemann zeta function. Ann. Prob. 25, 787–802.CrossRefGoogle Scholar
Fuh, C. D. (1997). Corrected diffusion approximations for ruin probabilities in a Markov random walk. Adv. Appl. Prob. 29, 695–712.Google Scholar
Fuh, C. D. (2003). SPRT and CUSUM in hidden Markov models. Ann. Statist. 31, 942–977.Google Scholar
Fuh, C. D. (2004). Uniform Markov renewal theory and ruin probabilities in Markov random walks. Ann. Appl. Prob. 14, 1202–1241.Google Scholar
Fuh, C. D. and Lai, T. L. (1998). Wald's equations, first passage times and moments of ladder variables in Markov random walks. J. Appl. Prob. 35, 566–580.Google Scholar
Fuh, C. D. and Lai, T. L. (2001). Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks. Adv. Appl. Prob. 33, 652–673.Google Scholar
Fuh, C. D. and Zhang, C. H. (2000). Poisson equation, moment inequalities and quick convergence for Markov random walks. Stoch. Process. Appl. 87, 53–67.Google Scholar
Glasserman, P. and Kou, S. (1995). Limits of first passage times to rare sets in regenerative processes. Ann. Appl. Prob. 5, 424–445.Google Scholar
Hoglund, T. (1991). The ruin problem for finite Markov chains. Ann. Prob. 19, 1298–1310.Google Scholar
Kartashov, N. V. (1996). Strong Stable Markov Chains. VSP, Utrecht.Google Scholar
Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Prob. 2, 355–386.Google Scholar
Klüppelberg, C. and Pergamenshchikov, S. M. (2003). Renewal theory for functionals of a Markov chain with compact state space. Ann. Prob. 31, 2270–2300.Google Scholar
Lalley, S. P. (1984). Limit theorems for first-passage times in linear and non-linear renewal theory. Adv. Appl. Prob. 16, 766–803.CrossRefGoogle Scholar
Lorden, G. (1970). On excess over the boundary. Ann. Math. Statist. 41, 520–527.Google Scholar
Lotov, V. I. (1996). On some boundary crossing problems for Gaussian random walks. Ann. Prob. 24, 2154–2171.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.CrossRefGoogle Scholar
Miller, H. D. (1962a). A matrix factorization problem in the theory of random variables defined on a finite Markov chain. Proc. Camb. Philios. Soc. 58, 268–285.Google Scholar
Miller, H. D. (1962b). Absorption probabilities for sums of random variables defined on a finite Markov chain. Proc. Camb. Philios. Soc. 58, 286–298.CrossRefGoogle Scholar
Ney, P. and Nummelin, E. (1987). Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Prob. 15, 561–592.Google Scholar
Siegmund, D. (1979). Corrected diffusion approximations in certain random walk problems. Adv. Appl. Prob. 11, 701–719.Google Scholar
Siegmund, D. (1982). Large deviations for boundary crossing probabilities. Ann. Prob. 10, 581–588.Google Scholar
Siegmund, D. (1985). Sequential Analysis. Springer, New York.Google Scholar
Siegmund, D. (1988). Approximate tail probabilities for the maxima of some random fields. Ann. Prob. 16, 487–501.Google Scholar