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Functional limit theorems for critical processes with immigration

Part of: Stochastic processes Markov processes Parametric inference

Published online by Cambridge University Press:  01 July 2016

I. Rahimov
I. Rahimov*
Affiliation:
King Fahd University of Petroleum and Minerals
*
∗ Postal address: Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Box 1339, Dhahran 31261, Saudi Arabia. Email address: rahimov@kfupm.edu.sa
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Abstract

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We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.

Keywords

Branching processimmigrationfunctionalmartingale limit theoremSkorokhod spaceleast-squares estimator

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 62F12: Asymptotic properties of estimators 60G99: None of the above, but in this section

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 4 , December 2007 , pp. 1054 - 1069
Copyright
Copyright © Applied Probability Trust 2007 

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