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Limit theorems and estimation theory for branching processes with an increasing random number of ancestors

Part of: Markov processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

J. P. Dion  and
N. M. Yanev
J. P. Dion*
Affiliation:
Université du Québec à Montréal
N. M. Yanev*
Affiliation:
Bulgarian Academy of Sciences
*
∗Postal address: Department of Mathematics, UQAM, C.P. 8888, Succ. Centre-Ville, Montreal, Quebec, H3C 3P8, Canada.
∗∗Postal address: Institute of Mathematics, Bulgarian Academy of Sciences, 8 G. Bontchev Str., 1113 Sofia, Bulgaria.
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Abstract

This paper deals with a Bienaymé-Galton-Watson process having a random number of ancestors. Its asymptotic properties are studied when both the number of ancestors and the number of generations tend to infinity. This yields consistent and asymptotically normal estimators of the mean and the offspring distribution of the process. By exhibiting a connection with the BGW process with immigration, all results can be transported to the immigration case, under an appropriate sampling scheme. A key feature of independent interest is a new limit theorem for sums of a random number of random variables, which extends the Gnedenko and Fahim (1969) transfer theorem.

Keywords

BRANCHING PROCESSESLIMIT THEOREMSESTIMATION THEORY;RANDOM SUMS OF RVIMMIGRATIONRANDOM NUMBER OF ANCESTORS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 62M05: Markov processes
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 309 - 327
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Partially supported by Bulgarian National Foundation for Scientific Investigations.

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