Theory of Probability and Mathematical Statistics
The asymptotic behavior of the distribution of Markov moments in time-inhomogeneous Markov chains and its application to a discrete Cramér-Lundberg model
M. V. Kartashov
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Abstract: 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 τ, 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 τ tends to infinity in the scheme of series, τ→∞. The asymptotic behavior of the ruin (killing) probabilities P(τ<∞)→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.
Keywords: Inhomogeneous discrete Markov chains, rare Markov moments, ruin probability, Cramér's risk model, analytical method
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