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The speed of extinction for some generalized jiřina processes

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Yuqiang Li
Yuqiang Li*
Affiliation:
East China Normal University
*
∗ Postal address: School of Finance and Statistics, East China Normal University, Shanghai 200241, P. R. China. Email address: [email protected]
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Abstract

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The speed of extinction for some generalized Jiřina processes {Xn} is discussed. We first discuss the geometric speed. Under some mild conditions, the results reveal that the sequence {cn}, where c does not equal the pseudo-drift parameter at x = 0, cannot estimate the speed of extinction accurately. Then the general case is studied. We determine a group of sufficient conditions such that Xn/cn, with a suitable constant cn, converges almost surely as n → ∞ to a proper, nondegenerate random variable. The main tools used in this paper are exponent martingales and stochastic growth models.

Keywords

Generalized Jiřina processextinctiongeometric series

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F15: Strong theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 576 - 599
Copyright
Copyright © Applied Probability Trust 2009 

References

Jiřina, M. (1960). Stochastic branching processes with continuous state space. Czech. Math. J. 83, 292–313.Google Scholar
Jiřina, M. (1966). Asymptotic behaviour of measure-valued branching processes. Rozpr. Česk. Akad. Věd. Řada Mat. Přir. Věd. 76, 1–55.Google Scholar
Kuster, P. (1985). Asymptotic growth of controlled Gaton–Watson processes. Ann. Prob. 13, 1157–1178.CrossRefGoogle Scholar
Li, Y. (2006). On a continuous-state population-size-dependent branching process and its extinction. J. Appl. Prob. 43, 195–207.CrossRefGoogle Scholar
Li, Y. (2009). Approximating nonlinear models of interest rates with branching processes. Acta Math. Sci., 29, 1–9 (in Chinese).Google Scholar
Li, Y. (2009). A weak limit theorem for generalized Jiřina processes. J. Appl. Prob. 46, 453–462.CrossRefGoogle Scholar
Li, Y. (2009). Limit theorems for generalized Jiřina processes. Statist. Prob. Lett. 79, 158–164.CrossRefGoogle Scholar
Seneta, E. and Vere-Jones, D. (1969). On a problem of M. Jiřina concerning continuous state branching processes. Czech. Math. J. 9, 277–283.CrossRefGoogle Scholar