On the maximal inequality
type: Item
instance of: scholarly article
title: On the maximal inequality
zbMATH Open document ID: 0886.60021
DOI: 10.1016/S0167-7152(96)00017-X
author: Qiying Wang
published in: Statistics \& Probability Letters
publication date: 1998-05-04T00:00:00Z
review text: Let \(X_1\), \(X_2, \dots, X_n,\dots\) be a sequence of real i.i.d. random variables, \(S_n: =\sum^n_{k=1} X_k\), \(\lambda>0\) a real number, \(h>0\) a non-decreasing, convex real function, then \[ P\Bigl(\sup_{k\geq n}S_k\geq\varepsilon\Bigr)\leq Eh(\lambda S_n)/h (\lambda \varepsilon n) \] for every real \(\varepsilon>0\) and every integer \(n\geq 1\). This is proved using a reverse-martingale inequality and a moment estimate. Corollaries sharpen among others Hoeffding's and Banjevic's inequalities.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 1031726
zbMATH Keywords: maximal inequality
zbMATH Keywords: reverse martingale
reviewed by: Heinrich Hering
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1031726