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Some limit theorems for positive recurrent branching Markov chains: I

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Krishna B. Athreya  and
Hye-Jeong Kang
Krishna B. Athreya*
Affiliation:
Iowa State University
Hye-Jeong Kang*
Affiliation:
Seoul National University
*
∗ Postal address: Departments of Mathematics and Statistics, Iowa State University, Ames, Iowa 50011, USA. Email address: [email protected]
∗∗ Postal address: Global Analysis Research Center, Department of Mathematics, Seoul National University, Seoul 151-742, Korea.
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Abstract

In this paper we consider a Galton-Watson process whose particles move according to a Markov chain with discrete state space. The Markov chain is assumed to be positive recurrent. We prove a law of large numbers for the empirical position distribution and also discuss the large deviation aspects of this convergence.

Keywords

Positive recurrent Markov chainsGalton-Watson processeslaw of large numberslarge deviation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 693 - 710
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Research supported in part by a National Science Foundation Grant #DMS 9204938.

References

Asmussen, S. (1987). Applied Probability and Queues. Wiley, Chichester.Google Scholar
Asmussen, S. and Kaplan, N. (1976). Branching random walks II. Stoch. Proc. Appl. 4, 15–31.Google Scholar
Athreya, K. B. (1994). Large deviation rates for branching processes: I Single type case. Ann. Appl. Prob. 4, 779–790.Google Scholar
Athreya, K. B. and Kang, H.-J. (1998). Some limit theorems for positive recurrent branching Markov chains: II. Adv. Appl. Prob. 30, 711–722.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. (1972). Branching Processes. Springer, New York.Google Scholar
Chow, Y. S. and Teicher, H. (1988). Probability Theory, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Ikeda, N., Nagasaway, M. and Watanabe, S. (1968). Branching Markov processes. I. J. Math. Kyoto Univ. 2, 233–278.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, London.Google Scholar
Kang, H-J. (1996a). Law of large numbers for branching Levy processes. Submitted.Google Scholar
Kang, H-J. (1996b). Central limit theorems for Bellman–Harris processes. Submitted.Google Scholar
Kesten, H. (1989). Supercritical branching processes with countably many types and the sizes of random Cantor sets. In Statistics and Mathematics Papers in Honor of Samuel Karlin, ed. Anderson, K. B. Athreya, Iglehart, D. L.. Academic Press, New York, pp. 108–121.Google Scholar
Kesten, H. and Stigum, B. P. (1967). Limit theorems for decomposable multidimensional Galton–Watson process. J. Math. Anal. Appl. 17, 309–338.Google Scholar
Kurtz, T. (1972). Inequalities for law of large numbers. Ann. Math. Statist. 43, 1874–1883.Google Scholar
Mode, C. J. (1971). Multitype Branching Processes. Elsevier, New York.Google Scholar
Royden, H. L. (1987). Real Analysis, 3rd edn. Macmillan Publishing Company, New York.Google Scholar
Schinazi, R. (1993). On multiple phase transitions for branching Markov chains. J. Statist. Phys. 71, 521–525.Google Scholar
Watanabe, S. (1965). On the branching process for Brownian particles with an absorbing boundary. J. Math. Kyoto Univ. 4, 185–198.Google Scholar