The Maclaurin inequality through the probabilistic lens

Abstract

In this paper we take a probabilistic look at Maclaurin's inequality, which is a refinement of the classical AM-GM inequality. In a natural randomized setting, we obtain limit theorems and show that a reverse inequality holds with high probability. The form of Maclaurin's inequality naturally relates it to U-statistics. More precisely, given $x_1, \ldots, x_n, p \in (0,\infty)$ and $k \in \mathbb{N}$ with $k \leq n$, let us define the quantity \[ S_{k, p}^{(n)} = \Big( \tbinom{n}{k}^{-1} \sum_{1 \leq i_1 < \ldots < i_k \leq n} x_{i_1}^p \cdots x_{i_k}^p \Big)^{1/(k p)}.\] Then as a consequence of the classical Maclaurin inequalities, we know that $S_{k_1}^{(n)} \geq S_{k_2}^{(n)}$ for $k_1 < k_2$. In the present article we consider the ratio \[ \mathcal{R}_{k_1, k_2, p}^{(n)} := \frac{S_{k_2, p}^{(n)}}{S_{k_1, p}^{(n)}}, \] evaluated at a random vector $(X_1, \ldots, X_n)$ sampled either from the normalized surface measure on the $\ell_p^n$-sphere or from a distribution generalizing both the uniform distribution on the $\ell_p^n$-ball and the cone measure on the $\ell_p^n$-sphere; by the Maclaurin inequality, we always have $\mathcal{R}_{k_1, k_2, p}^{(n)} \leq 1$. We derive central limit theorems for $\mathcal{R}_{k_1, k_2, p}^{(n)}$ and $\mathcal{R}_{k_1, n, p}^{(n)}$ as well as Berry--Esseen bounds and a moderate deviations principle for $\mathcal{R}_{k_1, n, p}^{(n)}$, keeping $k_1$, $k_2$ fixed, in order to quantify the set of points where $\mathcal{R}_{k_1, k_2, p}^{(n)} > c$ for $c \in (0, 1)$, i.e., where the Maclaurin inequality is reversed up to a factor. The present aricle partly generalizes results concerning the AM-GM inequality obtained by Kabluchko, Prochno, and Vysotsky (2020), Th\"ale (2021), and Kaufmann and Th\"ale (2023+).