Related papers: The Gaussian correlation inequality for centered c…

We consider the Gaussian correlation inequality for nonsymmetric convex sets. More precisely, if $A\subset\mathbb{R}^d$ is convex and the origin $0\in A$, then for any ball $B$ centered at the origin, it holds $\gamma_d(A\cap B)\geq…

The paper is to prove the Gaussian correlation conjecture stating that, under the standard Gaussian measure, the measure of the intersection of any two symmetric convex sets is greater than or equal to the product of their measures.…

We present in detail Thomas Royen's proof of the Gaussian correlation inequality which states that $\mu(K\cap L)\geq \mu(K)\mu(L)$ for any centered Gaussian measure $\mu$ on $R^d$ and symmetric convex sets $K,L$ in $R^d$.

The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the…

In this paper, we compare two variances of maxima of $N$ standard Gaussian random variables. One is a sequence of $N$ i.i.d. standard Gaussians, and the other one is $N$ standard Gaussians with covariances $\sigma_{1,2}=\rho \in(0,1)$ and…

Gaussian correlation conjecture states that the Gaussian measure of the intersection of two symmetric convex sets is greater or equal to the product of the measures.

We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure $\gamma$ on $\mathbb{R}^n$, stating…

We provide a sharp quantitative version of the Gaussian concentration inequality: for every $r>0$, the difference between the measure of the $r$-enlargement of a given set and the $r$-enlargement of a half-space controls the square of the…

This note presents families of inequalities for the Gaussian measure of convex sets which extend the recently proven Gaussian correlation inequality in various directions.

Motivated by the study of the propagation of convexity by semi-groups of stochastic differential equations and convex comparison between the distributions of solutions of two such equations, we study the comparison for the convex order…

We describe various sets of conditional independence relationships, sufficient for qualitatively comparing non-vanishing squared partial correlations of a Gaussian random vector. These sufficient conditions are satisfied by several…

A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures…

Let $\mu$ be a Gaussian measure (say, on ${\bf R}^n$) and let $K, L \subset {\bf R}^n$ be such that K is convex, $L$ is a "layer" (i.e. $L = \{x : a \leq < x,u > \leq b \}$ for some $a$, $b \in {\bf R}$ and $u \in {\bf R}^n$) and the…

The long-standing Gaussian product inequality (GPI) conjecture states that, for any centered $\mathbb{R}^n$-valued Gaussian random vector $(X_1, \dots, X_n)$ and any positive reals $\alpha_1, \dots, \alpha_n$, ${\bf…

Let $\prec$ be the product order on $\mathbb{R}^k$ and assume that $X_1,X_2,\ldots,X_n$ ($n\geq3$) are i.i.d. random vectors distributed uniformly in the unit hypercube $[0,1]^k$. Let $S$ be the (random) set of vectors in $\mathbb{R}^k$…

In this paper we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in $\mathbb{R}^{2}$. While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects…

We prove the 3-dimensional Gaussian product inequality, i.e., for any real-valued centered Gaussian random vector $(X,Y,Z)$ and $m\in \mathbb{N}$, it holds that…

The Gaussian product inequality is an important conjecture concerning the moments of Gaussian random vectors. While all attempts to prove the Gaussian product inequality in full generality have been unsuccessful to date, numerous partial…

We show that for some constant $\kappa>0$, any centered $\kappa$-subgaussian random variable is equal to the sum of three standard Gaussian random variables, confirming a conjecture of M. Talagrand. We also prove that given $\Lambda\geq 1$,…

This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and…