Mean and Minimum of Independent Random Variables

Abstract

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

arXiv comment18 pages, 3 figures