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An extension of Cramér's estimate for the absorption probability of a random walk

  • E. Arjas (a1) and T. P. Speed (a2)

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Consider a real-valued random walk

which is defined on a Markov chain {Xn: n ≥ 0} with countable state space I. We assume that a matrix Q(.) = (qij(.)) is given such that

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References

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(1)Cinlar, E.Markov renewal theory. Adv. Appl. Prob. 1 (1969), 123187.
(2)Cramér, H.On some questions connected with mathematical risk. Univ. California Publ. Statist. 2 (1954), 99124.
(3)Feller, W.An introduction to probability theory and its applications, 2nd ed., volume 2 (Wiley, New York, 1971).
(4)Keilson, J. and Wishart, D. M. G.A central limit theorem for processes defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 60 (1964), 547567.
(5)Kemeny, J. G., Snell, J. L. and Knapp, A. W.Denumerable Markov chains (Van Nostrand, Princeton, 1966).
(6)Kingman, J. F. C.A convexity property of positive matrices. Quart. J. Math. Oxford 12 (1961), 283284.
(7)Matthews, J. P. A study of processes associated with a finite Markov chain. Ph.D. thesis (1971), University of Sheffield (unpublished).
(8)Miller, H. D.A matrix factorization problem in the theory of random variables defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 58 (1962), 268285.
(9)Miller, H. D.Absorption probabilities for sums of random variables defined on a finite Markov chain. Proc. Cambridge Philos. Soc. 58 (1962), 286298.
(10)Presman, E.Factorization methods and boundary problems for sums of random variables given on Markov chains. Izv. Akad. Nauk USSR Ser. Mat. 33 (1969). (English translation. Amer. Math. Soc. (1971), 818852.)
(11)Vere-Jones, D.Ergodic properties of non-negative matrices, I. Pacific J. Math. 23 (1967), 601620.

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