On a contraction property of Bernoulli canonical processes
Volume 67 / 2019
Abstract
We give several results concerning suprema of canonical processes. The main theorem concerns a contraction property of Bernoulli canonical processes which generalizes the one proved by Talagrand (1993). It states that for independent Rademacher random variables $(\varepsilon_i)_{i\geq1}$ we can compare $\mathbf{E}\,\sup_{t\in T}\sum_{i\geq1}\varphi_{i}(t)\varepsilon_i$ with $\mathbf{E}\,\sup_{t\in T}\sum_{i=1}^{\infty}t_i\varepsilon_i$, where the function $\varphi=(\varphi_i)_{i\geq1}: T\rightarrow\ell^2$, $T\subset\ell^2$, satisfies certain conditions. Originally, it was assumed that each $\varphi_i$ is a contraction. We relax this assumption to comparability of Gaussian parts of increments: for all $s,t\in T$ and $p\ge 0$, $$ \inf_{|I^c|\le Cp}\sum_{i\in I}|\varphi_i(t)-\varphi_i(s)|^2\le C^2\inf_{|I^c|\le p}\sum_{i\in I}|t_i-s_i|^2, $$ where $C\ge 1$ is an absolute constant and $I\subset{\mathbb N}$, $I^c={\mathbb N}\setminus I$.
