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Stochastic stability of some state-dependent growth-collapse processes

Part of: Markov processes Stochastic processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Christian Y. Robert
Christian Y. Robert*
Affiliation:
CNAM and CREST
*
∗ Postal address: Centre de Recherche en Economie et Statistique, Timbre J320, 15 Boulevard Gabriel Peri, 92245 Malakoff Cedex, France. Email address: chrobert@ensae.fr
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Abstract

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In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

Keywords

Growth-collapse processMarkov chainlong-range dependenceheavy-tailed distribution

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60G10: Stationary processes 60G51: Processes with independent increments; L'evy processes 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 189 - 220
Copyright
Copyright © Applied Probability Trust 2007 

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