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