Mean and Minimum of Independent Random Variables

TL;DR

The paper investigates the asymptotic behavior of the minimum of two independent random variables conditioned on their sum, revealing that this probability tends to zero as the threshold grows, with extensions to multivariate cases.

Contribution

It establishes a new limit property for independent random variables and proves multivariate and weighted generalizations under identical distribution assumptions.

Findings

The probability that the minimum exceeds a threshold given the sum exceeds twice that threshold tends to zero.

The result holds for non-compactly supported independent random variables on [0,∞).

Multivariate and weighted generalizations are proved under identical distribution conditions.

Abstract

We show that any pair $X, Y$ of independent, non-compactly supported random variables on $[0,\infty)$ satisfies $\liminf_{m\to\infty} \mathbb{P}(\min(X,Y) >m \,| \,X+Y> 2m) =0$. We conjecture multi-variate and weighted generalizations of this result, and prove them under the additional assumption that the random variables are identically distributed.