On Hoeffding's inequalities.
type: Item
instance of: scholarly article
title: On Hoeffding's inequalities.
zbMATH Open document ID: 1062.60011
DOI: 10.1214/009117904000000360
published in: The Annals of Probability
publication date: 2004-09-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0410159
review text: Let \(X_j, 1 \leq j \leq n\), be bounded independent random variables and define \(M_n=X_1+\dots+X_n.\) The author proves that \(P(M_n \geq x) \leq c P(S_n \geq x)\) where \(c\) is an absolute constant and \(S_n= \varepsilon_1+\dots+\varepsilon_n\) is a sum of independent identically distributed Bernoulli random variables. The inequality holds for those real numbers \(x\) at which the function \(B(x)=P(S_n\geq x)\) has a downward jump. For other values of \(x,\) the inequality holds provided that the function \(B(x)=P(S_n \geq x)\) is linearly or log-linearly interpolated between adjacent jump points. The results are extended to martingales with bounded differences.
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 2100707
zbMATH Keywords: probabilities of large deviations
zbMATH Keywords: martingale
zbMATH Keywords: bounds for tail probabilities
zbMATH Keywords: inequalities
zbMATH Keywords: bounded differences and random variables
zbMATH Keywords: Hoeffding's inequalities
author: Vidmantas Bentkus
reviewed by: B. L. S. Prakasa Rao
MaRDI profile type: Publication
OpenAlex ID: W2037982184
arXiv ID: math/0410159
cites work: Q3708047
cites work: Q5671986
cites work: On the asymptotical behavior of the constant in the Berry-Esseen inequality
cites work: An inequality for large deviation probabilities of sums of bounded i. i. d. random variables
cites work: An inequality for tail probabilities of martingales with differences bounded from one side
cites work: On measure concentration for separately Lipschitz functions in product spaces
cites work: Estimates of the proximity of a distribution to the normal law
cites work: On conservative confidence intervals
cites work: Q4747394
cites work: A Note on Symmetric Bernoulli Random Variables
cites work: A probability inequality for linear combinations of bounded random variables
cites work: On the distribution of the number of successes in independent trials
cites work: A Topological Application of the Isoperimetric Inequality
cites work: On the Distribution of the Number of Successes in Independent Trials
cites work: Probability Inequalities for Sums of Bounded Random Variables
cites work: Q3234855
cites work: Q5534404
cites work: Some Results for Discrete Unimodality
cites work: Q4938929
cites work: Inequalities: theory of majorization and its applications
cites work: Q3496342
cites work: Q3715637
cites work: Q4730292
cites work: Asymptotic theory of finite dimensional normed spaces. With an appendix by M. Gromov: Isoperimetric inequalities in Riemannian manifolds
cites work: Sur quelques applications des fonctions convexes et concaves au sens de I. Schur
cites work: Some comments on deviation inequalities for infinitely divisible random vectors
cites work: Q4085018
cites work: Extremal probabilistic problems and Hotelling's \(T^ 2\) test under a symmetry condition
cites work: Q4387152
cites work: Q4263966
cites work: Q3395931
cites work: Concentration of measure and isoperimetric inequalities in product spaces
cites work: The missing factor in Hoeffding's inequalities
description: scientific article