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Monotone Stochastic Recursions and their Duals

Published online by Cambridge University Press:  27 July 2009

Søren Asmussen  and
Karl Sigman
Søren Asmussen
Affiliation:
Department of Mathematical Statistics, University of Lund, Box 118S-221 00 Lund, Sweden
Karl Sigman
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, 500 West 120th Street, New York, New York 10027–6699, USA
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Abstract

A duality is presented for real-valued stochastic sequences [Vn] defined by a general recursion of the form Vn+1 = f(Vn, Un), with [Un] a stationary driving sequence and f nonnegative, continuous, and monotone in its first variable. The duality is obtained by defining a dual function g of f, which if used recursively on the time reversal of [Un] defines a dual risk process. As a consequence, we prove that steady-state probabilities for Vn can always be expressed as transient probabilities of the dual risk process. The construction is related to duality of stochastically monotone Markov processes as studied by Siegmund (1976, The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes, Annals of Probability 4: 914–924). Our method of proof involves an elementary sample-path analysis. A variety of examples are given, including random walks with stationary increments and two reflecting barriers, reservoir models, autoregressive processes, and branching processes. Finally, general stability issues of the content process are dealt with by expressing them in terms of the dual risk process.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 1 , January 1996 , pp. 1 - 20
Copyright
Copyright © Cambridge University Press 1996

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