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A Nonsymmetric Correlation Inequality for Gaussian Measure

✍ Scribed by Stanislaw J. Szarek; Elisabeth Werner

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 182 KB
Volume: 68
Category: Article
ISSN: 0047-259X
DOI: 10.1006/jmva.1998.1784

No coin nor oath required. For personal study only.

✦ Synopsis

Let + be a Gaussian measure (say, on R n ) and let K, L R n be such that K is convex, L is a layer'' (i.e., L=[x: a (x, u) b] for some a, b # R and u # R n ), and the centers of mass (with respect to +) of K and L coincide. Then +(K & L) +(K) } +(L). This is motivated by the well-known positive correlation conjecture'' for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses the estimate 8(x)> 1&((8Â?) 1Â2 Â(3x+(x 2 +8) 1Â2 )) e &x 2 Â2 , x>&1, for the (standard) Gaussian cumulative distribution function, which is sharper than the classical inequality of Komatsu.

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