Journal of Applied Probability

Asymptotic behaviour of extinction probability of interacting branching collision processes

Anyue Chen, Junping Li, Yiqing Chen, and Dingxuan Zhou

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Abstract

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.

Article information

Source
J. Appl. Probab. Volume 51, Number 1 (2014), 219-234.

Dates
First available in Project Euclid: 25 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1395771425

Digital Object Identifier
doi:10.1239/jap/1395771425

Mathematical Reviews number (MathSciNet)
MR3189453

Zentralblatt MATH identifier
1291.60200

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Keywords
Markov branching process interacting branching collision process extinction probability asymptotic behaviour

Citation

Chen, Anyue; Li, Junping; Chen, Yiqing; Zhou, Dingxuan. Asymptotic behaviour of extinction probability of interacting branching collision processes. J. Appl. Probab. 51 (2014), no. 1, 219--234. doi:10.1239/jap/1395771425. https://projecteuclid.org/euclid.jap/1395771425.


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