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The xlogx condition for general branching processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Peter Olofsson
Peter Olofsson*
Affiliation:
Rice University
*
∗Postal address: Department of Statistics – MS 138, Rice University, 6100 Main Street, Houston, TX 77005–1892, USA. Email address: [email protected].
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Abstract

The xlogx condition is a fundamental criterion for the rate of growth of a general branching process, being equivalent to non-degeneracy of the limiting random variable. In this paper we adopt the ideas from Lyons, Pemantle and Peres (1995) to present a new proof of this well-known theorem. The idea is to compare the ordinary branching measure on the space of population trees with another measure, the size-biased measure.

Keywords

General branching processx log x conditionimmigration

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 537 - 544
Copyright
Copyright © Applied Probability Trust 1998 

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References

Athreya, K. B. (1997). Change of measures for Markov chains and the L log L theorem for branching processes. Technical Report #M97-9, Department of Mathematics, Iowa State University.Google Scholar
Athreya, K. B., and Ney, P. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Durrett, R. (1991). Probability: Theory and Examples. Wadsworth, Pacific Grove, CA.Google Scholar
Jagers, P. (1989). General branching processes as Markov fields. Stoch. Proc. Appl. 32, 183–212.CrossRefGoogle Scholar
Jagers, P., and Nerman, O. (1984). The growth and composition of branching populations. Adv. Appl. Prob. 16, 221–259.CrossRefGoogle Scholar
Lyons, R., Pemantle, R., and Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behaviour of branching processes. Ann. Prob. 3, 1125–1138.Google Scholar
Nerman, O. (1981). On the convergence of supercritical general (c-m-j) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.CrossRefGoogle Scholar
Olofsson, P. (1996). General branching processes with immigration. J. Appl. Prob. 33, 940–948.CrossRefGoogle Scholar