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


Author: M. V. Kartashov
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 92 (2015).
Journal: Theor. Probability and Math. Statist. 92 (2016), 37-58
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
DOI: https://doi.org/10.1090/tpms/981
Published electronically: August 10, 2016
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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 $ \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.


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Additional Information

M. V. Kartashov
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email: mkartashov@skif.com.ua

DOI: https://doi.org/10.1090/tpms/981
Keywords: Inhomogeneous discrete Markov chains, rare Markov moments, ruin probability, Cram\'er's risk model, analytical method
Received by editor(s): March 4, 2015
Published electronically: August 10, 2016
Article copyright: © Copyright 2016 American Mathematical Society