Domain of Attraction of the Quasistationary Distribution for Birth-and-Death Processes

Abstract

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, {aj(λC)}, 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→∞ℙM(X(t)=j∣ T>t)= aj(λC), 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<∞.