On the First Exit Time of a Nonnegative Markov Process Started at a Quasistationary Distribution
Moshe Pollak¹, Alexander G. Tartakovsky²
¹Statistics, The Hebrew University of Jerusalem, Jerusalem, Israel; ²Mathematics, The University of Southern California, Los Angeles, CA, United States

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_n→∞ P(M_n ≤ x | M₀ ≤ A, M₁ ≤ A, …, M_n ≤ A). Suppose that M₀ has distribution Q_A and define T_A^Q_A = min{ n | M_n > 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_A^Q_A to be stochastically nondecreasing in A.

Keywords: Quasistationary distribution; First exit time; Markov process; Changepoint problems

Biography: Moshe Pollak is Marcy Bogen Professor of Statistics at the Hebrew University of Jerusalem. His main line of research is in sequential statistical process control.