The Annals of Probability

On the Gaussian measure of the intersection

G. Schechtman, Th. Schlumprecht, and J. Zinn

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Abstract

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 measures. In this paper 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.

Article information

Source
Ann. Probab., Volume 26, Number 1 (1998), 346-357.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855422

Digital Object Identifier
doi:10.1214/aop/1022855422

Mathematical Reviews number (MathSciNet)
MR1617052

Zentralblatt MATH identifier
0936.60015

Subjects
Primary: 60E15: Inequalities; stochastic orderings 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]

Keywords
Gaussian measures correlation log-concavity

Citation

Schechtman, G.; Schlumprecht, Th.; Zinn, J. On the Gaussian measure of the intersection. Ann. Probab. 26 (1998), no. 1, 346--357. doi:10.1214/aop/1022855422. https://projecteuclid.org/euclid.aop/1022855422


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References

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