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Limit theorems for sums of a sequence of random variables defined on a Markov chain

Published online by Cambridge University Press:  14 July 2016

David B. Wolfson
David B. Wolfson*
Affiliation:
McGill University, Montreal
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Abstract

Let {(Jn, Xn), n ≧ 0} be the standard J–X process of Markov renewal theory. Suppose {Jn, n ≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that if converges in distribution, where an, bn > 0 (bn → ∞) are real constants, then the limit law F must be stable. Suppose Q(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn), n ≧ 0}. Then the n-fold convolution, Q∗n(bnx + anbn), converges in distribution to F(x)Π if and only if converges in distribution to F. Π is the matrix of stationary transition probabilities of {Jn, n ≧ 0}. Sufficient conditions on the Hi's are given for the convergence of the sequence of semi-Markov matrices to F(x)Π, where F is stable.

Keywords

STABLE LAWSSEMI-MARKOV PROCESSESSTRONG MIXING CONDITION
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 3 , September 1977 , pp. 614 - 620
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Chung, K. L. (1972) A Course in Probability Theory. Academic Press, New York.Google Scholar
[2] Fabens, A. J. and Neuts, M. F. (1970) The limiting distribution of the maximum term in a sequence of random variables defined on a Markov chain. J. Appl. Prob. 7, 754–760.CrossRefGoogle Scholar
[3] Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, Mass.Google Scholar
[4] Ibragimov, I. A. (1962) Some limit theorems for stationary processes. Theor. Prob. Appl. 7, 349–382.CrossRefGoogle Scholar
[5] Loève, M. (1963) Probability Theory. Van Nostrand, Princeton, New Jersey.Google Scholar
[6] O'Brien, G. L. (1974) Limit theorems for sums of chain dependent processes. J. Appl. Prob. 11, 582–587.CrossRefGoogle Scholar
[7] Ortega, J. ?. (1972) Numerical Analysis. A Second Course. Academic Press, New York.Google Scholar
[8] Ott, T. (1973) Infinite divisibility and stability of finite semi-Markov matrices. University of Rochester Report CSS 73–11, 31 July 1973.Google Scholar
[9] Resnick, S. I. and Neuts, M. F. (1970) Limit laws for maximum of a sequence of random variables defined on a Markov chain. Adv. Appl. Prob. 2, 323–343.CrossRefGoogle Scholar
[10] Serfozo, R. F. (1973) Weak convergence of superpositions of randomly selected partial sums. Ann. Prob. 1, 1044–1056.CrossRefGoogle Scholar
[11] Tucker, H. (1968) Convolutions of distributions attracted to stable laws. Ann. Math. Statist. 39, 1381–1390.CrossRefGoogle Scholar