Decoupling and domination inequalities with application to Wald's identity for martingales
The aim of the present paper is to show that combining the theory of sums of dominated random variables and the theory of decoupling inequalities [started in a paper by \textit{J. Zinn}, in: Probability in Banach spaces V. Lect. Notes Math. 1153, 453-457 (1985; Zbl 0571.60058)] can be useful in proving domination results involving dependent random variables and arbitrary stopping times. A typical result in this direction is the following. Let \((X_i)\subset L^p\) \((p\geq 1)\) be a sequence of random variable adapted to a filtration \(({\mathcal F}_i)\). Let \((Y_i)\subset L^p\) be a sequence of random variables conditionally independent given by \({\mathcal G}= \sigma (X_1,X_2, \dots)\). Let, for any \(t>0\), \(x\in R\) and \(i=1,2, \dots\) the following inequality be satisfied: \[ P\bigl(|X_i+x |> t\mid{\mathcal F}_{i-1}\bigr)\leq P\bigl(|Y_i+x|>t\mid{\mathcal G}\bigr)\text{ a.e.} \] Let \(T\) be a finite \({\mathcal G}\)-measurable stopping time. Then \[ E\left[\Phi \left(\left |\sum^T_{i=1} X_i\right|\right)\right]\leq k_pE\left[\Phi \left(\left |\sum^T_{i=1} Y_i\right|\right)\right] \] for all convex \(\Phi\in{\mathcal A}_p\). Here, \({\mathcal A}_p\) denotes the set of all non-decreasing continuous functions \(\Phi:[0,\infty [\to R\) such that \(\Phi(0)=0\) and \(\Phi(cx)\leq c^p \Phi(x)\) for all \(c\geq 2\) and \(x\geq 0\).
scientific article; zbMATH DE number 1796730