A refinement of the \v{S}id\'ak-Khatri inequality and a strong Gaussian correlation conjecture

TL;DR

This paper proves special cases of a strengthened Gaussian correlation conjecture and uses these results to refine the de1k-Khatri inequality, linking conjectural asymptotic behavior to finite-dimensional cases.

Contribution

It establishes specific cases of a strengthened Gaussian correlation conjecture and derives a refined de1k-Khatri inequality based on these cases.

Findings

Proved two special cases of the Gaussian correlation conjecture.

Showed that asymptotic validity implies finite-dimensional validity.

Refined the de1k-Khatri inequality using these special cases.

Abstract

We prove two special cases of a strengthened Gaussian correlation conjecture, due to Tehranchi, and show that if the conjecture holds asymptotically, it holds for any dimension. Additionally, we use these special cases to prove a refined version of the \v{S}id\'ak-Khatri inequality.