Some remarks on the Oleszkiewicz problem

TL;DR

This paper investigates how to verify when the expectation of the supremum of one Bernoulli process dominates another, providing a partial solution to the Oleszkiewicz conjecture on moment comparability in Banach spaces.

Contribution

It establishes a comparison result for Bernoulli processes analogous to Gaussian cases, addressing the Oleszkiewicz conjecture.

Findings

Proves a comparison theorem for Bernoulli processes

Provides partial evidence for the Oleszkiewicz conjecture

Extends Gaussian process comparison techniques to Bernoulli processes

Abstract

In this paper we study the question how to easily verify that the expectation of the supremum of a one canonical Bernoulli process dominates the same quantity for another process of this type. In the setting of Gaussian canonical processes it is known that such a comparison holds for contractions in the Euclidean distance. We do state and prove a similar result for Bernoulli processes. In particular we get a partial answer to the Oleszkiewicz conjecture about the comparability of weak and strong moments for type Bernoulli series in a Banach space.