A Nonsymmetric Correlation Inequality for Gaussian Measure |
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Authors: | Stanislaw J Szarek Elisabeth Werner |
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Institution: | a Case Western Reserve University;Université Pierre & Marie Curie, Paris, France;b Université de Lille, Villeneuve d 'Ascq, France |
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Abstract: | Letμbe a Gaussian measure (say, onRn) and letK,L Rnbe such thatKis convex,Lis a “layer” (i.e.,L={x: a![less-than-or-equals, slant less-than-or-equals, slant](http://www.sciencedirect.com/scidirimg/entities/2a7d.gif) x, u![right-pointing angle bracket right-pointing angle bracket](http://www.sciencedirect.com/scidirimg/entities/232a.gif) b} for somea, b Randu Rn), and the centers of mass (with respect toμ) ofKandLcoincide. Thenμ(K∩L) μ(K)·μ(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 estimateΦ(x)> 1−((8/π)1/2/(3x+(x2+8)1/2))e−x2/2,x>−1, for the (standard) Gaussian cumulative distribution function, which is sharper than the classical inequality of Komatsu. |
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Keywords: | Sidak's inequality correlation conjecture nonsymmetric correlation inequality Gaussian tail estimates |
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