Fluctuation limit of branching processes with immigration and estimation of the means

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in wh...

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Bibliographic Details
Published in:Advances in applied probability 2005-06, Vol.37 (2), p.523-538
Main Authors: Ispány, M., Pap, G., van Zuijlen, M. C. A.
Format: Article
Language:English
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Summary:We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2 , in contrast to both the subcritical case, in which it is n 1/2 , and the nearly critical case with positive limiting offspring variance, in which it is n .
ISSN:0001-8678
1475-6064
DOI:10.1017/S000186780000029X