Concentration of measure for products of Markov kernels and graph products via functional inequalities (Q2722655)

An inequality due to M. Talagrand asserts the following:NEWLINENEWLINENEWLINELet \(\nu_n\) be the \(n\)-fold product measure of \(\nu\) with \(d\nu(x)= e^{-|x|}/2 dx\), then for any Borel set \(A\subseteq \mathbb R^n\) with \(\nu_n(A)\geq 1/2\) and any \(r>0\) it follows that NEWLINE\[NEWLINE\nu_n(A+\sqrt{r} B_2+ r B_1)\geq 1-e^{-r/C} . \tag{1}NEWLINE\]NEWLINE Here \(B_1\) and \(B_2\) are the unit balls in \(\mathbb R^n\) w.r.t. the \(l_1\)- and \(l_2\)-norm, respectively, and \(C>0\) denotes some universal constant. In a recent paper \textit{S. G. Bobkov} and \textit{M. Ledoux} [Probab. Theory Relat. Fields 107, No. 3, 347-365 (1997; Zbl 0878.60014)] found some functional concentration inequality for \(\nu_n\) implying (1). The main aim of the present paper is to generalize this concentration inequality into various different directions. First the authors show how this concentration property of \(\nu_n\) follows from a discrete log-Sobolev type inequality. Then concentration of measure phenomena are derived for Markov kernels and graphs. The approach relies heavily on various notions and properties of the so-called discrete gradient.