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 λ_n,n≥0, and death coefficients μ_n,n≥0. If we define A=∑_n=1^∞ 1/λ_nπ_n and S=∑_n=1^∞ (1/λ_nπ_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_jxQ_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<∞.