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On the Gaussian measure of the intersection. (English) Zbl 0936.60015

Summary: The Gaussian correlation conjecture states that for any two symmetric, convex sets in \(n\)-dimensional space and for any centered, Gaussian measure an that space, the measure of the intersection is greater than or equal to the product of the measures. We obtain several results which substantiate this conjecture. For example, in the standard Gaussian case, we show there is a positive constant, \(c\), such that the conjecture is true if the two sets are in the Euclidean ball of radius \(c\sqrt n\). Further we show that if for every \(n\) the conjecture is true when the sets are in the Euclidean ball of radius \(\sqrt n\), then it is true in general. Our most concrete result is that the conjecture is true if the two sets are (arbitrary) centered ellipsoids.

MSC:

60E15 Inequalities; stochastic orderings
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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