The asymptotic behavior of the distribution of Markov moments in time-inhomogeneous Markov chains and its application to a discrete Cramér–Lundberg model

A time-inhomogeneous discrete Markov chain and a family of substochastic matrices $(Q_{s})$ subordinated to (bounded from above by) the one-step transition probabilities $(P_{s})$ of the chain are considered. The Markov moment $\tau$, the killing moment for 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 some sense, so that the moment $\tau$ tends to infinity in the scheme of series, $\tau \rightarrow \infty$. The asymptotic behavior of the ruin (killing) probabilities $\mathsf {P}(\tau <\infty )\rightarrow 0$ is found. To prove this result, we assume that a condition like the transiency of the chain holds (as in the well-known Cramér–Lundberg theorem). Some applications are also discussed.