On a contraction property of Bernoulli canonical processes

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 (εi)i≥1 we can compare E suptϵT Σi≥1 φi(t) εi with E suptϵT Σi=1∞ ti εi, where the function φ = (φi)i≥1: T → l2, T ⊂ l2, satisfies certain conditions. Originally, it was assumed that each φi is a contraction. We relax this assumption to comparability of Gaussian parts of increments: for all s, t ϵ T and p ≥ 0, [formula], where C ≥ 1 is an absolute constant and I ⊂ N, Ic = N \ I.

---

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).