The asymptotic behavior of rare Markov moments defined on time inhomogeneous Markov chains
Author:
M. V. Kartashov
Translated by:
N. Semenov
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
Full-text PDF Free Access
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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.
References
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References
- W. Doeblin, Expose de la theorie des chaines simples constantes de Markov a un nomber fini d’estats, Mathematique de l’Union Interbalkanique 2 (1938), 77–105.
- I. I. Gikhman and A. V. Skorokhod, Theory of Stochastic Processes, “Nauka”, Moscow, 1973; English transl., vol. I, Springer–Verlag, New York–Heidelberg–Berlin, 1974; vol. II, 1975; vol. III, 1979.
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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
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