A Sharp Estimate for Probability Distributions

TL;DR

This paper provides a sharp lower bound on the probability that, given the sum of two independent non-compactly supported continuous random variables exceeds a threshold, at least one of them is small, refining previous asymptotic results.

Contribution

It establishes a universal lower bound involving the median and the essential supremum of the distribution, improving understanding of the tail behavior of sums of continuous distributions.

Findings

The bound is sharp up to constants.

The logarithmic dependence on median and supremum is necessary.

The result fails for compactly supported distributions.

Abstract

We consider absolutely continuous probability distributions $f(x)dx$ on $\mathbb{R}_{\geq 0}$. A result of Feldheim and Feldheim shows, among other things, that if the distribution is not compactly supported, then there exist $z > 0$ such that most events in $\left\{X + Y \geq 2z\right\}$ are comprised of a 'small' term satisfying $\min(X,Y) \leq z$ and a 'large' term satisfying $\max(X,Y) \geq z$ (as opposed to two 'large' terms that are both larger than $z$) $$ \limsup_{z \rightarrow \infty}~ \mathbb{P}\left( \min(X,Y) \leq z | X+Y \geq 2z\right) = 1.$$ The result fails if the distribution is compactly supported. We prove $$\sup_{z > 0 } ~\mathbb{P}\left( \min(X,Y) \leq z | X+Y \geq 2z\right) \geq \frac{1}{24 + 8\log_2{( med(X) \|f\|_{L^{\infty}})}},$$ where $med(X)$ denotes the median. Interestingly, the logarithm is necessary and the result is sharp up to constants; we also discuss some open problems.