QUANTITATIVE AND QUALITATIVE LIMITS FOR EXPONENTIAL ASYMPTOTICS OF HITTING TIMES FOR BIRTH-AND-DEATH CHAINS IN A SCHEME OF SERIES

摘要

We consider a time-homogeneous discrete birth-and-death Markov chain (X_t) and investigate the asymptotics of the hitting time τ_n = inf（t ≥ 1： X_t ≥ n） as well as the chain position before this time in the scheme of series as n → oo. In our case, one-step probabilities of the chain vary simultaneously with n. The proofs are based on the explicit two-sided inequalities with numerical bounds for the survival probability P（τ_n > t）. These inequalities can be used also for the pre-limit finitetime schemes. We have applied the results obtained for constructing the uniform asymptotic representations of the corresponding risk function.