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

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 λ n ,n≥0, and death coefficients μ n ,n≥0. If we define A=∑ n=1 ∞ 1/λ n π n and S=∑ n=1 ∞ (1/λ n π n )∑ i=n+1 ∞ π i , where {π n ,n≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A=∞ and S<∞, then λ C >0 and there is precisely one quasistationary distribution, namely, {a j (λ C )}, where λ C is the decay parameter of {X(t),t≥0} in C={1,2,...} and a j (x)≡μ1 -1π j xQ j (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={m i , i=1,2,...}, we have lim t→∞ℙ M (X(t)=j∣ T>t)= a j (λ C ), 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<∞.