Quantitative and qualitative limits for exponential asymptotics of hitting times for birth-and-death chains in a scheme of series

We consider a time-homogeneous discrete birth-and-death Markov chain $(X_{t})$ and investigate the asymptotics of the hitting time $\tau _n=\inf (t\geq 1\colon X_{t}\geq n)$ as well as the chain position before this time in the scheme of series as $n\to \infty$. 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 $\mathsf {P}(\tau _n>t)$. These inequalities can be used also for the pre-limit finite-time schemes. We have applied the results obtained for constructing the uniform asymptotic representations of the corresponding risk function.