Decoupling and domination inequalities with application to Wald's identity for martingales
type: Item
instance of: scholarly article
title: Decoupling and domination inequalities with application to Wald's identity for martingales
zbMATH Open document ID: 1008.60040
DOI: 10.1016/S0167-7152(02)00042-1
author: Victor H. de la Peña
author: Ingrid-Mona Zamfirescu
published in: Statistics \& Probability Letters
publication date: 2002-09-05T00:00:00Z
review text: 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 \(x\geq 0\).
reviewed by: Klaus Schürger
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 1796730
zbMATH Keywords: dominated random variables
zbMATH Keywords: decoupling inequalities
zbMATH Keywords: stopping times
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1796730