Decoupling and domination inequalities with application to Wald's identity for martingales

Victor H. de la Peña and Ingrid-Mona Zamfirescu

Statistics & Probability Letters, 2002, vol. 57, issue 2, 157-170

Abstract: The aim of this paper is to show how decoupling techniques can be useful in proving domination results involving dependent random variables and arbitrary stopping times. Let {Xi} be a sequence of random variables adapted to , an increasing sequence of [sigma]-fields and set . Assume that for all t>0, (0.1) where {Yi} is a sequence of conditionally independent given . A case of particular importance is the one when the {Yi} is independent of in which case the conditioning drops from (0.1). Then, for all stopping times T adapted to , E maxj[less-than-or-equals, slant]T [summation operator]i=1jXip[less-than-or-equals, slant]CpE[summation operator]i=1TYip whenever {Xi} is a mean-zero martingale difference sequence and p[greater-or-equal, slanted]1 or it is a sequence of non-negative or conditionally symmetric random variables and p>0. As a consequence of this, we obtain a sharp extension of Wald's equation for randomly stopped sums of i.i.d. random variables to the case of sums of the form [summation operator]i=kTXi-k+1...Xi where the mean-zero martingale sequence {Xi} satisfies (0.1).

Keywords: Martingales Moving averages Decoupling Domination inequalities Stopping times; Wald's equation (search for similar items in EconPapers)
Date: 2002
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