zbMATH — the first resource for mathematics

A new look at independence. (English) Zbl 0858.60019
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”.

60E15 Inequalities; stochastic orderings
28A35 Measures and integrals in product spaces
Full Text: DOI