Weak dimension-free concentration of measure
Let \(\mu\) be a probability measure on the Borel \(\sigma\)-algebra \({\mathcal B}(\mathbb{R})\), let \(F\) denote its cumulative distribution function, set \(F^{ -1} (p)=\inf \{x\in\mathbb{R}: F(x)\geq p\}\), \(0<p<1\), and define \(U(x)=F^{-1} ((1+ e^{-x})^{-1})\), \(x\in\mathbb{R}\). Further, let \(\mu^n\) be the \(n\)-fold product of \(\mu\) with itself, assume \(\mathbb{R}^n\) is equipped with the supremum distance \(\rho \), and put \(A^h=\{x\in\mathbb{R}^n:\rho(x,a)\leq h\) for some \(a\in A\}\) whenever \(A \subset\mathbb{R}^n\) and \(h>0\). Consider also the values \(R_h^{(n)}(p)= \inf\{\mu^n (A^h): A\in{\mathcal B}(\mathbb{R}^n)\), \(\mu^n(A)\geq p\}\), \(h>0\), \(0<p<1\). The main result is as follows. For \(0<p<1\), the following statements are equivalent: (a) There exists \(h>0\) such that \(\inf_nR_h^{(n)} (p)>p\). (b) There exist \(\delta> 0\) and \(c>0\) such that \(F(x)-F(x- \delta)\geq cF(x) (1-F(x))\), \(x\in\mathbb{R}\). (c) The function \(U\) has finite modulus of continuity. In this case, for \(h>0\), setting \(h^*=\sup \{|U(x)-U(y) |:x,y \in\mathbb{R}\), \(|x-y|\leq h\}\), one has \(\inf_nR_{h^*}^{(n)} (p)\geq p/(p+(1-p)e^{-h})\). This last inequality is then expressed in more probabilistic language in terms of a sequence \(\xi_1,\xi_2, \dots\) of i.i.d. random variables defined on some probability space. Concentration properties of \(\max\{\xi_1, \dots, \xi_n\}\) and \(\min\{ \xi_1, \dots, \xi_n\}\) are also studied.
scientific article; zbMATH DE number 1529278