Abstract
Some results for stopped random walks are extended to the Markov renewal setup where the random walk is driven by a Harris recurrent Markov chain. Some interesting applications are given; for example, a generalization of the alternating renewal process.
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REFERENCES
Alsmeyer, G. (1988). Second-order approximations for certain stopped sums in extended renewal theory, Advances in Applied Probability, 20, 391–410.
Alsmeyer, G. (1994). On the Markov renewal theorem, Stochastic Process. Appl., 50, 37–56.
Anscombe, F. J. (1952). Large-sample theory of sequential estimation, Proceedings of Cambridge Philosophical Society, 48, 600–607.
Asmussen, S. (1987). Applied Probability and Queues, Wiley, New York.
Athreya, K. B. and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains, Trans. Amer. Math. Soc., 245, 493–501.
Athreya, K. B., McDonald, D. and Ney, P. (1978). Limit theorems for semi-Markov processes and renewal theory for Markov chains, Ann. Probab., 6, 788–797.
Boshuizen, F. A. and Gouweleeuw, J. M. (1993). General optimal stopping theorems for semi-Markov processes, Advances in Applied Probability, 25, 825–846.
Gut, A. (1988). Stopped Random Walks, Springer, New York.
Gut, A. (1990). Cumulative shock models, Advances in Applied Probability, 22, 504–507.
Gut, A. (1992). First passage times of perturbed random walks, Sequential Anal., 11, 149–179.
Gut, A. and Ahlberg, P. (1981). On the theory of chromatography based upon renewal theory and a central limit theorem for randomly indexed partial sums of random variables, Chemica Scripta, 18, 248–255.
Gut, A. and Janson, S. (1983). The limiting behaviour of certain stopped sums and some applications, Scand. J. Statist., 10, 281–292.
Horváth, L. (1986). Strong approximations of renewal processes and their applications, Acta Math. Hungar., 47, 13–28.
Janson, S. (1983). Renewal theory for m-dependent variables, Ann. Probab., 11, 558–568.
Kao, E. P. C. (1973). Optimal replacement rules when changes of state are semi-Markovian, Oper. Res., 21, 1231–1249.
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Alsmeyer, G., Gut, A. Limit Theorems for Stopped Functionals of Markov Renewal Processes. Annals of the Institute of Statistical Mathematics 51, 369–382 (1999). https://doi.org/10.1023/A:1003822611607
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DOI: https://doi.org/10.1023/A:1003822611607