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 λn, n ≥ 0 and death coefficients μn, n ≥ 0. We recall that the series A = ∑∞ n=1 1 λnπn and the series S = ∑∞ n=1 1 λnπn ∑∞ i=n+1 πi, where {πn, n ≥ 1} is the potential coefficients. It is well-known fact (see van Doorn [13]) that if the A = ∞ and S < ∞, then λc > 0 and there is precisely one quasi-stationary distribution, viz. {aj(λc)}, where λc is the decay parameter of {X(t), t ≥ 0} in C and aj(x) ≡ μ−1 1 πjxQj(x), j = 1, 2, · · · . In this paper, we prove that there is a unique quasi-stationary distribution that attracts all initial distributions supports in C = {1, 2, · · · }, if and only if the birth-death process {X(t), t ≥ 0} satisfies both A =∞ and S <∞.