A new look at independence
type: Item
instance of: scholarly article
title: A new look at independence
zbMATH Open document ID: 0858.60019
author: Michel Talagrand
published in: The Annals of Probability
publication date: 1997-04-01T00:00:00Z
review text: Let \(X\) be the product of \(N\) probability spaces \((\Omega, \Sigma, \mu)\) and let \(P\) be the product measure \(\mu^{\otimes N}\) on \(X\). The distance \(d\) on \(X\) is defined by \(d(x,y) = \text{card}\{1 \leq i \leq N: x_i \neq y_i\}\), where \(x = (x_i; 1\leq i \leq N)\) and \(y = (y_i; 1\leq i \leq N)\). Given a set \(A \subset X\), consider its enlargement \(A_t :=\{x\in X: d(x,A)\leq t\}\) for \(t>0\). Then the bound \(1-P(A_t)\leq 2\text{exp} \{-t^2/N\}\) holds whenever \(P(A) \geq 1/2\). It is an example of what is called as the concentration of measure phenomenon. According to the author: ``The present paper represents my best attempt to explain in the simplest way I can achieve what this is all about, without ever doing anything technical''.
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 28A35
zbMATH DE Number: 926201
zbMATH Keywords: exponential inequalities
zbMATH Keywords: concentration of measure
zbMATH Keywords: product spaces
reviewed by: Rimas Norvaiša
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1042644705
OpenAlex ID: W2022392475
Wikidata QID: Q94752553
DOI: 10.1214/AOP/1042644705
description: scientific article; zbMATH DE number 926201