Consider some random variables $ X_1 $ , , , $ X_n $ they are independent and distributed identically to CDF $ F (x) $and some random variables $ Y_1. , , Y_n $ which are independent and identical to CDF distributed $ G (y) $, Suppose that everyone $ X_i $ is independent of everyone $ Y_i $,

These pairs of random variables give random sums. $ X_1 + U_1 $ , , , $ X_n + U_n $, What is the probability that the first random sum is the highest? That's clear $ 1 / n $ through symmetry. Equally clear is the likelihood that the first random sum is the highest **Fewer** when $ 1 / n $ if I assume that the first random sum falls below a certain limit. This is,

$$ P (X_1 + U_1 geq X_j + U_j hspace {0,1cm} for all i hspace {0,1cm} | hspace {0,1cm} U_i leq u) leq 1 / n $$

Presumably this is true with strict inequality, though $ n> 1 $, Is that true? I've spent a lot of time showing this, and would appreciate any help, ideas, or hints.

Thank you in advance!