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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: yqli@stat.ecnu.edu.cn
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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

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