On a contraction property of Bernoulli canonical processes

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 $$\mathbb{E}$\sup_{t\in T}\sum_{i\geq1}\varphi_{i}(t)\varepsilon_i$with$$\mathbb{E}$\sup_{t\in T}\sum_{i\geq1}t_i\varepsilon_i$,whereafunction$\varphi=(\varphi_i)_{i\geq1}: \ell^2\supset T\rightarrow\ell^2$, 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 $s,t\in T$, $p\ge 0$ $$\underset{\mathrm{\mid }{I}^c\mathrm{\mid }\le Cp}{\inf }\sum _{i\in I}\mathrm{\mid }{\phi }_i(t)-{\phi }_i(s){\mathrm{\mid }}^2\le C^2\underset{\mathrm{\mid }{I}^c\mathrm{\mid }\le p}{\inf }\sum _{i\in I}\mathrm{\mid }{t}_i-s_i{\mathrm{\mid }}^2,$$ where $C\ge 1$ is an absolute constant and $I\subset \mathbb{N}$, $I^c=\mathbb{N}\mathrm{\backslash }I$.