Advances in Applied Probability

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

M. Ispány, G. Pap, and M. C. A. van Zuijlen

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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 n3/2, in contrast to both the subcritical case, in which it is n1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

Article information

Adv. in Appl. Probab. Volume 37, Number 2 (2005), 523-538.

First available in Project Euclid: 15 June 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J60: Diffusion processes [See also 58J65] 62M05: Markov processes: estimation

Conditional least-squares estimator fluctuation limit nearly critical Galton-Watson branching process with immigration Ornstein-Uhlenbeck-type process


Ispány, M.; Pap, G.; van Zuijlen, M. C. A. Fluctuation limit of branching processes with immigration and estimation of the means. Adv. in Appl. Probab. 37 (2005), no. 2, 523--538. doi:10.1239/aap/1118858637.

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