The mixing advantage is less than 2

TL;DR

This paper establishes a sharp upper bound on the expected maximum of independent non-negative random variables based on their individual maxima expectations, providing insights into the limitations of the mixing advantage.

Contribution

It introduces a new tight inequality relating the expected maximum to individual maxima expectations, enhancing understanding of the mixing advantage in probability theory.

Findings

The derived bound is sharp and can be approached arbitrarily closely.

Provides comparison results related to the expected maximum.

Enhances theoretical understanding of the mixing advantage.

Abstract

Corresponding to $n$ independent non-negative random variables $X_1,...,X_n$, are values $M_1,...,M_n$, where each $M_i$ is the expected value of the maximum of $n$ independent copies of $X_i$. We obtain an upper bound to the expected value of the maximum of $X_1,...,X_n$ in terms of $M_1,...,M_n$. This inequality is sharp in the sense that the quantity and its bound can be made as close to each other as we want. We also present related comparison results.