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Extension of Wald's first lemma to Markov processes

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

George V. Moustakides
George V. Moustakides*
Affiliation:
University of Patras
*
∗Postal address: Department of Computer Engineering and Informatics, University of Patras, 26 500 Patras, Greece. Email address: moustaki@cti.gr.
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Abstract

Let ξ0,ξ1,ξ2,… be a homogeneous Markov process and let Sn denote the partial sum Sn = θ(ξ1) + … + θ(ξn), where θ(ξ) is a scalar nonlinearity. If N is a stopping time with 𝔼N < ∞ and the Markov process satisfies certain ergodicity properties, we then show that 𝔼SN = [limn→∞𝔼θ(ξn)]𝔼N + 𝔼ω(ξ0) − 𝔼ω(ξN). The function ω(ξ) is a well defined scalar nonlinearity directly related to θ(ξ) through a Poisson integral equation, with the characteristic that ω(ξ) becomes zero in the i.i.d. case. Consequently our result constitutes an extension to Wald's first lemma for the case of Markov processes. We also show that, when 𝔼N → ∞, the correction term is negligible as compared to 𝔼N in the sense that 𝔼ω(ξ0) − 𝔼ω(ξN) = o(𝔼N).

Keywords

Wald's lemmastopping times

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 48 - 59
Copyright
Copyright © Applied Probability Trust 1999 

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