## Abstract

In this paper we take a probabilistic look at Maclaurin’s inequality, which is a refinement of the classical AM–GM mean inequality. In a natural randomized setting, we obtain limit theorems and show that a reverse inequality holds with high probability. By its definition, Maclaurin’s inequality is naturally related to U-statistics. More precisely, given ${x}_{1},\dots ,{x}_{n},p\in (0,\mathrm{\infty})$ and $k\in \mathbb{N}$ with $k\le n$, let us define the quantity

$${S}_{k,p}^{(n)}={({\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}^{-1}\sum _{1\le {i}_{1}<\cdots <{i}_{k}\le n}{x}_{{i}_{1}}^{p}\cdots {x}_{{i}_{k}}^{p})}^{1\u2215(kp)}.$$

Then as a consequence of the classical Maclaurin inequalities, we know that ${S}_{{k}_{1},p}^{(n)}\ge {S}_{{k}_{2},p}^{(n)}$ 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 $({X}_{1},\dots ,{X}_{n})$ 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 ${\mathcal{R}}_{{k}_{1},{k}_{2},p}^{(n)}\le 1$. 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), Thäle (2021), and Kaufmann and Thäle (2023+).

## Funding Statement

LF was supported by the Austrian Science Fund (FWF), projects P-32405 and P-35322. JP is supported by the German Research Foundation (DFG) under project 516672205 and by the Austrian Science Fund (FWF) under project P-32405.

## Citation

Lorenz Frühwirth. Michael Juhos. Joscha Prochno. "The Maclaurin inequality through the probabilistic lens." Electron. J. Probab. 29 1 - 33, 2024. https://doi.org/10.1214/24-EJP1165

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