Mean and Minimum

Let X and Y be two unbounded positive independent random variables. Write Min_m for the probability of the event {min(X,Y) > m} and Mean_m for that of the event {(X+Y)/2 > m}. We show that the limit inferior of Min_m / Mean_m is always 0 (as m approaches infinity), regardless of the distributions of X and Y. We view this statement as a universal anti-concentration result, and discuss several implications. The proof is elementary but involved, relying on comparison to the ”nearest“ log-concave measure. We also provide a multiple-variables, weighted variant of this result in the i.i.d. case and pose a conjectured general result encompassing this phenomenon. Joint work with Ohad Feldheim