On the Expectation of the First Exit Time of a Nonnegative Markov Process Started at a Quasistationary Distribution

Moshe Pollak; Alexander Tartakovsky

paper

On the Expectation of the First Exit Time of a Nonnegative Markov Process Started at a Quasistationary Distribution

arXiv:1006.0965

Abstract

Let {M_n}_{n\ge 0} $b e an o nn e g a t i v e M a r k o v p r ocess w i t h s t a t i o na r y t r an s i t i o n p r o babi l i t i es . T h e q u a s i s t a t i o na r y d i s t r ib u t i o n sr e f er r e d t o in t hi s n o t e a r eo f t h e f or m Q_{A} (x) = l i m_{n \to \infty} P (M_{n} \leq x ∣ M_{0} \leq A, M_{1} \leq A, ..., M_{n} \leq A) . S u pp ose t ha t$ M_0 $ha s d i s t r ib u t i o n$ \Qb_A $an dd e f in e T_{A}^{Q_{A}} = min {n ∣ M_{n} > A, n \geq 1}, t h e f i r s tt im e w h e n M_{n} e x cee d s A . W e p r o v i d es u f f i c i e n t co n d i t i o n s f or E T_{A}^{Q_{A}}$ to be an increasing function of A.

6 pages

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