On a contraction property of Bernoulli canonical processes

Abstract

In this paper we improve Bernoulli comparison. The result works for independent Rademacher random variables $({\epsilon }_i{)}_{i\ge 1}$ and states that we can compare $\mathbb{E}{\sup }_{t\in T}{\sum }_{i\ge 1}{\phi }_i(t){\epsilon }_i$ with $\mathbb{E}{\sup }_{t\in T}{\sum }_{i\ge 1}{t}_i{\epsilon }_i$, where a function $\phi =({\phi }_i{)}_{i\ge 1}:{\ell }^2\supset T\to {\ell }^2$, satisfies certain conditions. Originally, it is assumed that each of ${\phi }_i$ is a contraction. We relax this assumption towards comparison of Gaussian parts of increments, which can be described in the following way. For all $s,t\in T$, $p\ge 0$ $$\underset{|I^c|\le Cp}{\inf }\sum _{i\in I}|{\phi }_i(t)-{\phi }_i(s){|}^2\le C^2\underset{|I^c|\le p}{\inf }\sum _{i\in I}|t_i-s_i{|}^2,$$ where $C\ge 1$ is an absolute constant and $I\subset \mathbb{N}$, $I^c=\mathbb{N}\backslash I$.

18 pages