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Martingale central limit theorems and asymptotic estimation theory for multitype branching processes

Published online by Cambridge University Press:  01 July 2016

Søren Asmussen  and
Niels Keiding
Søren Asmussen
Affiliation:
University of Copenhagen
Niels Keiding
Affiliation:
University of Copenhagen
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Abstract

This paper considers the problem of estimating the growth rate ρ of a p-type Galton–Watson process {Zn}. To this end, a general approach of possible independent interest to central limit theorems for discrete-time branching processes is developed. The idea is to adapt martingale central limit theory to martingale difference triangular arrays indexed by the set of all individuals ever alive. Iterated logarithm laws are derived by similar methods. Asymptotic distribution results and the a.s. asymptotic behaviour are derived for a maximum likelihood estimator based upon all parent–offspring combinations in a given number N of generations, and for the estimator which depends on the total generation sizes only.

Keywords

GALTON–WATSON PROCESSMULTITYPELAW OF THE ITERATED LOGARITHMCENTRAL LIMIT THEOREMMARTINGALELINEAR FUNCTIONALESTIMATION THEORYGROWTH RATEMULTIPHASE BIRTH PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 1 , March 1978 , pp. 109 - 129
Copyright
Copyright © Applied Probability Trust 1978 

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