Results related to the Gaussian product inequality conjecture for mixed-sign exponents in arbitrary dimension

TL;DR

This paper proves that a form of the Gaussian product inequality holds for any combination of positive and negative exponents across any dimension, and provides a conditional quantitative lower bound.

Contribution

It extends the Gaussian product inequality to mixed-sign exponents in arbitrary dimensions and offers a conditional quantitative lower bound based on the conjecture.

Findings

GPI holds for mixed-sign exponents in any dimension

A general quantitative lower bound is established

Results build on previous two- and higher-dimensional cases

Abstract

This note establishes that the opposite Gaussian product inequality (GPI) of the type proved by Russell & Sun (2022a) in two dimensions, and partially extended to higher dimensions by Zhou et al. (2024), continues to hold for an arbitrary mix of positive and negative exponents. A general quantitative lower bound is also obtained conditionally on the GPI conjecture being true.