Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-17T05:21:38.073Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Parametric inference Markov processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

M. Ispány ,
G. Pap  and
M. C. A. van Zuijlen
M. Ispány*
Affiliation:
University of Debrecen
G. Pap*
Affiliation:
University of Debrecen
M. C. A. van Zuijlen*
Affiliation:
Radboud University Nijmegen
*
∗ Postal address: Faculty of Informatics, University of Debrecen, Pf. 12, H-4010 Debrecen, Hungary.
∗ Postal address: Faculty of Informatics, University of Debrecen, Pf. 12, H-4010 Debrecen, Hungary.
∗∗∗∗ Postal address: Faculty of Mathematics, Radboud University Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands. Email address: m.vanzuijlen@science.ru.nl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Conditional least-squares estimatorfluctuation limitnearly critical Galton-Watson branching process with immigrationOrnstein-Uhlenbeck-type process

MSC classification

Primary: 62F12: Asymptotic properties of estimators
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J60: Diffusion processes 62M05: Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 523 - 538
Copyright
Copyright © Applied Probability Trust 2005 

References

Aliev, S. A. (1985). A limit theorem for Galton–Watson branching processes with immigration. Ukrain. Mat. Zh. 37, 656–659.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1968). Markov Processes. John Wiley, New York.Google Scholar
Grimvall, A. (1974). On the convergence of sequences of branching processes. Ann. Prob. 2, 1027–1045.Google Scholar
Heyde, C. C. and Seneta, E. (1972). Estimation theory for growth and immigration rates in a multiplicative process. J. Appl. Prob. 9, 235–256.CrossRefGoogle Scholar
Heyde, C. C. and Seneta, E. (1974). Notes on ‘Estimation theory for growth and immigration rates in a multiplicative process’. J. Appl. Prob. 11, 572–577.Google Scholar
Ispány, M., Pap, G. and van Zuijlen, M. C. A. (2003). Asymptotic behaviour of estimators of the parameters of nearly unstable INAR(1) models. In Foundations of Statistical Inference, eds Haitovsky, Y., Lerche, H. R. and Ritov, Y., Physica, Heidelberg, pp. 193–204.Google Scholar
Ispány, 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
Jacod, J. and Shiryayev, A. N. (2003). Limit Theorems for Stochastic Processes. Springer, Berlin.CrossRefGoogle Scholar
Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 36–54.Google Scholar
Klimko, L. A. and Nelson, P. I. (1978). On conditional least squares estimation for stochastic processes. Ann. Statist. 6, 629–642.Google Scholar
Lamperti, J. (1967). The limit of a sequence of branching processes. Z. Wahrscheinlichkeitsth. 7, 271–288.CrossRefGoogle Scholar
Li, Z.-H. (2000). Ornstein–Uhlenbeck type processes and branching processes with immigration. J. Appl. Prob. 37, 627–634.Google Scholar
Quine, M. P. (1976). Asymptotic results for estimators in a subcritical branching process with immigration. Ann. Prob. 4, 319–325.CrossRefGoogle Scholar
Sriram, T. N. (1994). Invalidity of bootstrap for critical branching process with immigration. Ann. Statist. 22, 1013–1023.CrossRefGoogle Scholar
Wei, C. Z. and Winnicki, J. (1989). Some asymptotic results for the branching process with immigration. Stoch. Process. Appl. 31, 261–282.CrossRefGoogle Scholar
Wei, C. Z. and Winnicki, J. (1990). Estimation of the means in the branching process with immigration. Ann. Statist. 18, 1757–1773.CrossRefGoogle Scholar