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Two limit theorems for ergodically regenerated stochastic processes

Published online by Cambridge University Press:  24 October 2008

Joseph Kupka
Joseph Kupka
Affiliation:
Monash University, Clayton, Victoria 3168, Australia
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Abstract

There is a subtle difference between the recurrent event of Feller and the regenerative phenomenon of Kingman: The former regenerates an entire ambient process, whereas the latter regenerates only itself. This paper generalizes Feller's definition to a discrete regenerative phenomenon E in association with an arbitrary discrete-time stochastic process X. Two limit theorems of a general character are proved for the process X when it is regenerated by an ergodic phenomenon E. The first implies that X becomes strictly stationary at a uniform rate which is determined solely by the asymptotic behaviour of E. The second is essentially a mixing convergence theorem which implies the asymptotic independence of regenerating and regenerated events. Applications include (1) the outright independence of the regenerating events and the regenerated events which occur ‘at the end of time’, and (2) the identification of the transient features of X with the null sets of the stationary limiting probability. Numerous open questions are posed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 3 , May 1983 , pp. 519 - 536
Copyright
Copyright © Cambridge Philosophical Society 1983

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