Journal of Applied Probability

Domain of attraction of the quasistationary distribution for birth-and-death processes

Hanjun Zhang and Yixia Zhu

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We consider a birth–death process {X(t),t≥0} on the positive integers for which the origin is an absorbing state with birth coefficients λn,n≥0, and death coefficients μn,n≥0. If we define A=∑n=1 1/λnπn and S=∑n=1 (1/λnπn)∑i=n+1 πi, where {πn,n≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A=∞ and S<∞, then λC>0 and there is precisely one quasistationary distribution, namely, {ajC)}, where λC is the decay parameter of {X(t),t≥0} in C={1,2,...} and aj(x)≡μ1-1πjxQj(x), j=1,2,... . In this paper we prove that there is a unique quasistationary distribution that attracts all initial distributions supported in C, if and only if the birth–death process {X(t),t≥0} satisfies bothA=∞ and S<∞. That is, for any probability measure M={mi, i=1,2,...}, we have limt\→∞M(X(t)=jT>t)= ajC), j=1,2,..., where T=inf{t≥0 : X(t)=0} is the extinction time of {X(t),t≥0} if and only if the birth–death process {X(t),t≥0} satisfies both A=∞ and S<∞.

Article information

J. Appl. Probab., Volume 50, Number 1 (2013), 114-126.

First available in Project Euclid: 20 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Domain of attraction quasistationary distribution birth-and-death process orthogonal polynomial duality


Zhang, Hanjun; Zhu, Yixia. Domain of attraction of the quasistationary distribution for birth-and-death processes. J. Appl. Probab. 50 (2013), no. 1, 114--126. doi:10.1239/jap/1363784428.

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