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 = { m i , i = 1 , 2 , … } be a probability distribution with ∑ i i m i < ∞ , then l i m t → ∞ P M ( X ( t ) = j | T > t ) = ν j , where ν j is a geometric distribution with parameter λ / μ . The first method acquires the results by computing directly while the second method bases on the theory of probability generating function.