A simple proof of the majorizing measure theorem
Let \(X=\{X_ t; t\in T\}\) be a centered Gaussian process with a parameter set \(T\) equipped with and bounded in the canonical metric \(d\). The author [Acta Math. 159, No. 1-2, 99-149 (1987; Zbl 0712.60044)] proved that if \(X\) is bounded with probability one, then there exists a probability measure \(m\) on \((T,d)\) such that for some universal constant \(K\) \[ \sup_{t\in T}\int_ 0^{\infty}\sqrt{\log (1/m(\{s\in T: d(s,t)\leq \epsilon\}))} d\epsilon \leq K E\sup_{t\in T}X_ t. \] The first proof of this statement was quite difficult and was based on properties of ultrametric spaces. In the paper under review a new proof is given. Now the proof relies upon Sudakov's minoration, the concentration of measure phenomenon, and a simple construction.
scientific article; zbMATH DE number 56875
scientific article