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Stability index for products of random transformations

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

P. H. Baxendale  and
R. Z. Khasminskii
P. H. Baxendale*
Affiliation:
University of Southern California
R. Z. Khasminskii*
Affiliation:
Wayne State University
*
∗ Postal address: Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA. Email address: baxendale@math.usc.edu
∗∗ Postal address: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.
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Abstract

The paper considers stability and instability properties of the Markov chain generated by the composition of an i.i.d. sequence of random transformations. The transformations are assumed to be either linear mappings or else mappings which can be well approximated near 0 by linear mappings. The main results concern the risk probabilities that the Markov chain enters or exits certain balls centered at 0. An application is given to the probability of extinction in a model from population dynamics.

Keywords

Lyapunov exponentstability indexmoment stabilityrandom matrix productsrisk theoryreliabilitypopulation dynamicsextinction probability

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G42: Martingales with discrete parameter 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998 , pp. 968 - 988
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Research supported in part by Office of Naval Research contracts N00014-91-J-1526 and N00014-96-1-0413.

Research supported in part by Office of Naval Research contracts N00014-93-1-0936 and N00014-95-1-0793.

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