The asymptotic behavior of rare Markov moments defined on time inhomogeneous Markov chains

We consider a time-inhomogeneous discrete Markov chain and a family of substochastic matrices $(Q_{s})$ subordinated to (or those that do not exceed) the one-step transition matrices $(P_{s})$ of the chain. The Markov moment $\tau$ (the so called killing moment) defined on the chain with transition matrices $(Q_{s})$ is connected to the family $(Q_{s})$. We assume that $P_{s}$ and $Q_{s}$ are close in the uniform metric, that is, $\tau \rightarrow \infty$ in the scheme of series. The asymptotic behavior of the ruin (killing) probability ${\mathsf P}(\tau >n)$ is studied as $n\rightarrow \infty$ under the assumption that the perturbations $P_{s}-Q_{s}$ are uniformly negligible with respect to time $s$. Some applications are also considered.