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Functional limit theorems for critical processes with immigration

Part of: Parametric inference Stochastic processes Markov processes

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

References

Aliev, S. A. (1985). A limit theorem for Galton–Watson branching processes with immigration. Ukrain. Mat. Zh. 37, 656–659.Google Scholar
Alzaid, A. A. and Al-Osh, M. (1990). An integer-valued pth order autoregressive (INAR(p)) process. J. Appl. Prob. 27, 314–324.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopaedia Math. Appl. 27). Cambridge University Press.CrossRefGoogle Scholar
Dion, J. P., Gauthier, G. and Latour, A. (1995). Branching processes with immigration and integer-valued time series. Serdica Math. J. 21, 123–136.Google Scholar
Feller, W. (1951). Diffusion processes in genetics. In Proc. Second Berkeley Symp. Math. Statist. Prob., ed. Neyman, P., University of California Press, Berkeley, CA, pp. 227–246.Google Scholar
Franke, J. and Seligmann, T. (1993). Conditional maximum likelihood estimates for INAR(1) processes and their application to modelling epileptic seizure counts. In Developments in Time Series Analysis, ed. Rao, T. S., Chapman and Hall, London, pp. 310–330.CrossRefGoogle Scholar
Ispàny, M., Pap, G. and Van Zuijlen, M. C. A. (2003). Asymptotic inference for nearly unstable INAR(1) models. J. Appl. Prob. 40, 750–765.Google Scholar
Ispàny, M., Pap, G. and Van Zuijlen, M. C. A. (2005). Fluctuation limit of branching processes with immigration and estimation of the means. Adv. Appl. Prob. 37, 523–538.Google Scholar
Ispàny, M., Pap, G. and Van Zuijlen, M. C. A. (2006). Critical branching mechanisms with immigration and Ornstein–Uhlenbeck type diffusions. Acta Sci. Math. (Szeged) 71, 821–850.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes. Springer, Berlin.Google Scholar
Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 36–54.Google Scholar
Lamperti, J. (1967a). Limiting distributions for branching processes. In Proc. Fifth Berkeley Symp. Math. Statist. Prob. University of California Press, Berkeley, CA, pp. 225–241.Google Scholar
Lamperti, J. (1967b). The limit of a sequence of branching processes. Z. Wahrscheinlichkeitsth. 7, 271–288.Google Scholar
Lindvall, T. (1972). Convergence of critical Galton–Watson branching processes. J. Appl. Prob. 9, 445–450.Google Scholar
Lindvall, T. (1974). Limit theorems for functionals of certain Galton–Watson branching processes. Adv. Appl. Prob. 6, 309–327.Google Scholar
Rahimov, I. (1995). Random Sums and Branching Stochastic Processes (Lecture Notes Statist. 96). Springer, New York.Google Scholar
Sriram, T. N. (1994). Invalidity of bootstrap for critical branching processes with immigration. Ann. Statist. 22, 1013–1023.Google Scholar
Wei, C. Z. and Winnicki, J. (1989). Some asymptotic results for branching processes with immigration. Stoch. Process. Appl. 31, 261–282.Google Scholar