Convex combinations of random variables stochastically dominate the parent for a new class of heavy-tailed distributions

Abstract

Stochastic dominance of a random variable by a convex combination of its independent copies has recently been shown to hold within the relatively narrow class of distributions with concave odds function, and later extended to broader families of distributions. A simple consequence of this surprising result is that the sample mean can be stochastically larger than the underlying random variable. We show that a key property for this stochastic dominance result to hold is the subadditivity of the cumulative distribution function of the reciprocal of the random variable of interest, referred to as the inverted distribution. By studying relations and inclusions between the different classes for which the stochastic dominance was proved to hold, we show that our new class can significantly enlarge the applicability of the result, providing a relatively mild sufficient condition.