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

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).