The Gaussian correlation conjecture is true

Probability and Statistics 1 Minute

You need to know: Euclidean space {\mathbb R}^n, origin in {\mathbb R}^n, convex set in {\mathbb R}^n, centrally symmetric set in {\mathbb R}^n (set symmetric with respect to the origin), integration in {\mathbb R}^n, n \times n matrix, matrix multiplication, determinant \text{det}(A) of a matrix A, inverse A^{-1} of a matrix A with \text{det}(A)\neq 0, symmetric matrix, transpose A^T of matrix A.

Background: A symmetric n\times n matrix \Sigma is called positive definite, if x^T\Sigma x > 0 for all non-zero x \in {\mathbb R}^n. Given a positive definite n\times n matrix \Sigma, let f:{\mathbb R}^n \to {\mathbb R} be a function given by f(x)=(2\pi)^{-n/2}\text{det}(\Sigma)^{-1/2}\exp(-\frac{1}{2}x^T\Sigma^{-1} x), x\in {\mathbb R}^n. For a subset A\subset{\mathbb R}^n, let \mu(A)=\int_A f(x)dx. \mu is called an n-dimensional Gaussian probability measure on {\mathbb R}^n, centred at the origin.

The Theorem: On 5th August 2014, Thomas Royen submitted to arxiv and Far East Journal of Theoretical Statistics a paper in which he proved that inequality \mu(A \cap B) \geq \mu(A) \cdot \mu(B) holds for all convex and centrally symmetric sets A,B \subset {\mathbb R}^n.

Short context: The statement of the Theorem was known as Gaussian correlation conjecture, and was an important conjecture in probability theory. A special case of the conjecture was formulated by Dunnett and Sobel in 1955, the general case was stated in 1972. Despite simple-looking formulation and efforts of many researchers, the conjecture remained open until 2014. The Theorem proves the conjecture, and it is now called Gaussian correlation inequality.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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