An Exponential Inequality for Symmetric Random Variables

TL;DR

This paper establishes a new exponential inequality for sums of independent symmetric random variables, providing bounds that relate the sum's tail probability to the sum of squares, useful for probabilistic analysis.

Contribution

It introduces a novel exponential inequality for symmetric i.i.d. random variables, linking tail probabilities to quadratic sums, which was not previously known.

Findings

The inequality bounds the probability of large sums given quadratic constraints.

The result applies to symmetric i.i.d. variables, broadening existing tail bounds.

It offers a new tool for probabilistic inequalities in symmetric random settings.

Abstract

We prove the following exponential inequality: Let $n\geq 1$ and let $X_1,...,X_n$ be $n$ independent identically distributed symmetric real-valued random variables. For any $x,y>0$, we have \[\mathbb{P}\big({X_1+...+X_n}\geq x,\, {X_1^2+...+X_n^2}\leq y\big)< \exp(-\frac{x^2}{2y})\,.\]