A general class of exponential inequalities for martingales and ratios
type: Item
instance of: scholarly article
title: A general class of exponential inequalities for martingales and ratios
zbMATH Open document ID: 0942.60004
author: Victor H. de la Peña
published in: The Annals of Probability
publication date: 1999-11-09T00:00:00Z
review text: The author derives several new exponential inequalities for a martingale difference sequence \((d_i,F_i)\), which satisfies either \(E[|d_j|^k\mid F_{j-1}]\leq (k!/2)\sigma^2_j c^{k-2}\) or \(P(|d_j|\leq c\mid F_{j-1})= 1\), for \(k>2\), \(0< c<\infty\), where \(\sigma^2_j= E[d^2_j\mid F_{j-1}]\). For example, put \(V^2_n= \sum^n_{j=1} \sigma^2_j\) and \(M_m= \sum^n_{j=1} d_j\), and then for all \(x,y>0\), \[ P(M_n\geq x, V^2_n\text{ for some }n)\leq \exp\Biggl\{-{x^2\over 2(y+ cx)}\Biggr\}. \] The proof of the above and related results is based on decoupling techniques introduced by \textit{S. Kwapien} and \textit{W. A. Woyczynski} [in: Almost everywhere convergence, 237-265 (1989; Zbl 0693.60033)] and further developed by the author and others. The author gives also results for continuous time square integrable martingales. He also shows that in the special case of conditionally symmetric random variables the integrability conditions on the sequence \(d_j\) can be relaxed.
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 1359640
zbMATH Keywords: exponential martingale inequalities
zbMATH Keywords: decoupling inequalities
reviewed by: Esko Valkeila
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677271
OpenAlex ID: W1971031270
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DOI: 10.1214/AOP/1022677271
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description: scientific article; zbMATH DE number 1359640