Sharpening the probabilistic Arithmetic-Geometric Mean Inequality

TL;DR

This paper refines large deviation estimates for the probabilistic p-generalized arithmetic-geometric mean inequality in high-dimensional $ ext{l}_p^n$-balls, providing precise asymptotic probability bounds for inequality improvements or reversals.

Contribution

It sharpens existing large deviation results for the inequality, offering asymptotically exact estimates on the probability of improvement or reversal up to a constant.

Findings

Provides concrete asymptotic probability bounds for inequality improvements.

Sharpens large deviation results in the spirit of Bahadur and Ranga Rao.

Offers precise estimates on the likelihood of the inequality being improvable or reversible.

Abstract

We consider the $p$-generalized arithmetic-geometric mean inequality for vectors chosen randomly from the $\ell_p^n$-ball in $\mathbb{R}^n$. In this setting the inequality can be improved or reversed up to a respective scalar constant with high probability, and central limit theorems and large deviation results with respect to this constant have been shown. We sharpen these large deviation results in the spirit of Bahadur and Ranga Rao, thereby providing concrete and asymptotically exact estimates on a non-logarithmic scale for the probability of the inequality being improvable or reversible up to a constant, respectively.