On a contraction property of Bernoulli canonical processes

Witold Bednorz ¹   ; Rafał Martynek ¹

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 2, pp. 187-209

Résumé

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 $I\subset{\mathbb N}$, $I^c={\mathbb N}\setminus I$.

Détail

DOI : 10.4064/ba8111-4-2019

Keywords: several results concerning suprema canonical processes main theorem concerns contraction property bernoulli canonical processes which generalizes proved talagrand states independent rademacher random variables varepsilon geq compare mathbf sup sum geq varphi varepsilon mathbf sup sum infty varepsilon where function varphi varphi geq rightarrow ell subset ell satisfies certain conditions originally assumed each varphi contraction relax assumption comparability gaussian parts increments inf sum varphi varphi inf sum i s where absolute constant subset mathbb mathbb setminus