A note on optimal probability lower bounds for centered random variables

TL;DR

This paper derives optimal lower bounds for the probabilities that centered random variables are non-negative, improving existing results and applying these bounds to second order Rademacher chaos.

Contribution

It introduces new optimal probability lower bounds for centered random variables based on their moments, advancing prior research.

Findings

Bounds are proven to be optimal.

Improved probability estimates over previous literature.

Applications to second order Rademacher chaos.

Abstract

In this note we obtain lower bounds for $\P(\xi\geq 0)$ and $\P(\xi>0)$ under assumptions on the moments of a centered random variable $\xi$. The obtained estimates are shown to be optimal and improve results from the literature. The results are applied to obtain probability lower bounds for second order Rademacher chaos.