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March 2014 Asymptotic behaviour of extinction probability of interacting branching collision processes
Anyue Chen, Junping Li, Yiqing Chen, Dingxuan Zhou
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J. Appl. Probab. 51(1): 219-234 (March 2014). DOI: 10.1239/jap/1395771425


Although the exact expressions for the extinction probabilities of the Interacting Branching Collision Processes (IBCP) were very recently given by Chen et al. [4], some of these expressions are very complicated; hence, useful information regarding asymptotic behaviour, for example, is harder to obtain. Also, these exact expressions take very different forms for different cases and thus seem lacking in homogeneity. In this paper, we show that the asymptotic behaviour of these extremely complicated and tangled expressions for extinction probabilities of IBCP follows an elegant and homogenous power law which takes a very simple form. In fact, we are able to show that if the extinction is not certain then the extinction probabilities {an} follow an harmonious and simple asymptotic law of anknρcn as n → ∞, where k and α are two constants, ρc is the unique positive zero of the C(s), and C(s) is the generating function of the infinitesimal collision rates. Moreover, the interesting and important quantity α takes a very simple and uniform form which could be interpreted as the `spectrum', ranging from -∞ to +∞, of the interaction between the two components of branching and collision of the IBCP.


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Anyue Chen. Junping Li. Yiqing Chen. Dingxuan Zhou. "Asymptotic behaviour of extinction probability of interacting branching collision processes." J. Appl. Probab. 51 (1) 219 - 234, March 2014.


Published: March 2014
First available in Project Euclid: 25 March 2014

zbMATH: 1291.60200
MathSciNet: MR3189453
Digital Object Identifier: 10.1239/jap/1395771425

Primary: 60J27
Secondary: 60J35

Rights: Copyright © 2014 Applied Probability Trust


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Vol.51 • No. 1 • March 2014
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