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

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 offsprin...

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Bibliographic Details
Published in:Journal of applied probability 1997-06, Vol.34 (2), p.309-327
Main Authors: Dion, J. P., Yanev, N. M.
Format: Article
Language:English
Subjects:
Estimation theory
Estimators
Immigration
Infinity
Mathematical models
Mathematical theorems
Operations research
Probability
Random numbers
Random variables
Statistical inferences
Statistical theories
Studies
Tin
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Summary: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.
ISSN:0021-9002
1475-6072
DOI:10.2307/3215372