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A particular case of correlation inequality for the Gaussian measure. (English) Zbl 0962.28013
The paper deals with the following conjecture, which is due to S. Das Gupta, I. Olkin, L. J. Savage, M. L. Eaton, M. Perlman and M. Sobel [Proc. 6th Berkely Symp. Math. Statist. Probab., Univ. Calif. 1970, 2, 241-265 (1972; Zbl 0253.60021)]. Let $$A$$ and $$B$$ be two symmetric convex sets; if $$\mu$$ is a centered Gaussian measure on $$\mathbb{R}^n$$, could we say that $$\mu(A\cap B)\geq \mu(A)\mu(B)$$?
In the paper, the conjecture is proved if $$A$$ is a centered ellipsoid and $$B$$ is simply symmetric. Two proofs are given, the first one is based on the log-concavity result of A. Prékopa [Acta Sci. Math. 34, 335-343 (1973; Zbl 0264.90038)], and the second one relies on the comparison of semigroups. As a corollary the authors obtain a comparison of the moments of a measure with an even, log-concave density with the moments of $$\mu$$.

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