On a contraction property of Bernoulli canonical processes

Abstract

In this paper we improve Bernoulli comparison. The result works for independent Rademacher random variables $(\varepsilon_i)_{i\geq1}$ and states that we can compare $\mathbb{E}\sup_{t\in T}\sum_{i\geq1}\varphi_{i}(t)\varepsilon_i$ with $\mathbb{E}\sup_{t\in T}\sum_{i\geq1}t_i\varepsilon_i$, where a function $\varphi=(\varphi_i)_{i\geq1}: \ell^2\supset T\rightarrow\ell^2$, satisfies certain conditions. Originally, it is assumed that each of $\varphi_i$ is a contraction. We relax this assumption towards comparison of Gaussian parts of increments, which can be described in the following way. For all $s,t\in T$, $p\geq 0$ $$ \inf_{|I^c|\leq Cp}\sum_{i\in I}|\varphi_i(t)-\varphi_i(s)|^2\leq C^2\inf_{|I^c|\leq p}\sum_{i\in I}|t_i-s_i|^2, $$ where $C\geq 1$ is an absolute constant and $I\subset\mathbb{N}$, $I^c=\mathbb{N}\backslash I$.