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## On a contraction property of Bernoulli canonical processes

### Volume 67 / 2019

Bulletin Polish Acad. Sci. Math. 67 (2019), 187-209 MSC: 60G15, 60G17,60G50. DOI: 10.4064/ba8111-4-2019 Published online: 23 July 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 $I\subset{\mathbb N}$, $I^c={\mathbb N}\setminus I$.

#### Authors

• Witold BednorzInstitute of Mathematics
University of Warsaw
Banacha 2
02-097 Warszawa, Poland
e-mail
• Rafał MartynekInstitute of Mathematics
University of Warsaw
Banacha 2
02-097 Warszawa, Poland
e-mail

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