On Weak Tail Domination of Random Vectors

TL;DR

This paper investigates the concept of weak tail domination in random vectors, demonstrating that under certain regularity conditions, it implies strong tail domination, linking to the Bernoulli conjecture.

Contribution

It establishes conditions under which weak tail domination leads to strong tail domination, connecting to a major open problem in probability theory.

Findings

Weak tail domination implies strong tail domination under regularity conditions.

A positive answer to Oleszkiewicz's question would follow from the Bernoulli conjecture.

The work links tail behavior of random vectors to fundamental conjectures in probability.

Abstract

Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture.