On the first exit time of a nonnegative markov process started at a quasistationary distribution

Let {M_n}_n≥0 be a nonnegative time-homogeneous Markov process. The quasistationary distributions referred to in this note are of the form Q_A(x) = lim_nn→∞ P(M_n ≤ x | M₀ ≥ A, M₁ ≥ A, . . . , M_n ≥ A). Suppose that M₀ has distribution Q_A, and define T ^QA_A = min{n | Mn > A, n ≥ 1}, the first time when M_n exceeds A. We provide sufficient conditions for Q_A(x) to be nonincreasing in A (for fixed x) and for T^QA_A to be stochastically nondecreasing in A.