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Stochastic Sequences with a Regenerative Structure that May Depend Both on the Future and on the Past

Part of: Stochastic processes Limit theorems Special processes Markov processes

Published online by Cambridge University Press:  04 January 2016

Sergey Foss  and
Stan Zachary
Sergey Foss*
Affiliation:
Heriot-Watt University and Sobolev Institute of Mathematics
Stan Zachary*
Affiliation:
Heriot-Watt University
*
∗ Postal address: School of Mathematics and Computer Sciences and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.
∗ Postal address: School of Mathematics and Computer Sciences and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.
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Abstract

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Many regenerative arguments in stochastic processes use random times which are akin to stopping times, but which are determined by the future as well as the past behaviour of the process of interest. Such arguments based on ‘conditioning on the future’ are usually developed in an ad-hoc way in the context of the application under consideration, thereby obscuring the underlying structure. In this paper we give a simple, unified, and more general treatment of such conditioning theory. We further give a number of novel applications to various particle system models, in particular to various flavours of contact processes and to infinite-bin models. We give a number of new results for existing and new models. We further make connections with the theory of Harris ergodicity.

Keywords

Regenerative processbreak pointdependence on the future and on the pastcontact processinfinite-bin modelHarris ergodicity

MSC classification

Primary: 60K05: Renewal theory 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60K40: Other physical applications of random processes 60J05: Discrete-time Markov processes on general state spaces 60G40: Stopping times; optimal stopping problems; gambling theory 60F99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 1083 - 1110
Copyright
© Applied Probability Trust 

Footnotes

Research of both authors was partially supported by EPSRC grant EP/I017054/1.

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