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Theory of Probability and Mathematical Statistics

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


Author: M. V. Kartashov
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 88 (2013).
Journal: Theor. Probability and Math. Statist. 88 (2014), 109-121
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
DOI: https://doi.org/10.1090/S0094-9000-2014-00922-9
Published electronically: July 24, 2014
MathSciNet review: 3112638
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Abstract: 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.


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Additional Information

M. V. Kartashov
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Email: nkartashov@skif.com.ua

DOI: https://doi.org/10.1090/S0094-9000-2014-00922-9
Keywords: Inhomogeneous discrete Markov chains, rare Markov moments, ruin probability, coupling method
Received by editor(s): October 4, 2012
Published electronically: July 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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