Domain of attraction of the quasi-stationary distribution for the linear birth and death process

We obtain a sufficient condition for an initial distribution to be in the domain of attraction of the quasi-stationary distribution for a linear birth and death process by two methods, that is, let M={mi,i=1,2,…} be a probability distribution with ∑iimi<∞∑iimi<∞, thenlimt→∞PM(X(t)=j|T>t)=νj, where νjνj is a geometric distribution with parameter λ/μλ/μ. The first method acquires the results by computing directly while the second method bases on the theory of probability generating function.