On Gaussian Correlation Inequalities for Rectangles and Symmetric Sets

Abstract

Pitt's conjecture (1977) that P(A ∩ B) ≥ P(A)P(B) under the N_n (0, I_n) distribution of X, where A, B are symmetric convex sets in IRⁿ still lacks a complete proof. This note establishes that the above result is true when A is a symmetric rectangle while B is any symmetric convex set, where A, B ∈ IRⁿ. We give two different proofs of the result, the key component in the first one being a recent result by Hargé (1999). The second proof, on the other hand, is based on a rather old result of Šidák (1968), dating back a period before Pitt's conjecture.