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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A time-inhomogeneous discrete Markov chain and a family of substochastic matrices subordinated to (bounded from above by) the one-step transition probabilities of the chain are considered. The Markov moment , the killing moment for the chain with transition matrices , is connected to the family . We assume that and are close in some sense, so that the moment tends to infinity in the scheme of series, . The asymptotic behavior of the ruin (killing) probabilities 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