论文信息 - A Nonsymmetric Correlation Inequality for Gaussian Measure

A Nonsymmetric Correlation Inequality for Gaussian Measure

Let?be a Gaussian measure (say, onRn) and letK,L?Rnbe such thatKis convex,Lis a “layer” (i.e.,L={x:a??x,u??b} for somea,b?Randu?Rn), and the centers of mass (with respect to?) ofKandLcoincide. 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?(x)> 1?((8/?)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.

Elisabeth M. Werner | Stanislaw J. Szarek | S. Szarek | E. Werner

[1] Z. Šidák. Rectangular Confidence Regions for the Means of Multivariate Normal Distributions , 1967 .

[2] Mary Beth Ruskai,et al. A one-dimensional model for many-electron atoms in extremely strong magnetic fields: maximum negative ionization , 1999 .

[3] Gideon Schechtman,et al. ON THE GAUSSIAN MEASURE OF THE INTERSECTION , 1998 .

[4] L. Pitt. A Gaussian Correlation Inequality for Symmetric Convex Sets , 1977 .

[5] H. McKean,et al. Diffusion processes and their sample paths , 1996 .

[6] Richard J. Bagby,et al. CALCULATING NORMAL PROBABILITIES , 1995 .

[7] C. Borell. A Gaussian correlation inequality for certain bodies in ℝn , 1981 .

[8] C. G. Khatri,et al. On Certain Inequalities for Normal Distributions and their Applications to Simultaneous Confidence Bounds , 1967 .

[9] A. Ehrhard. Symétrisation dans l'espace de Gauss. , 1983 .

[10] Yaozhong Hu,et al. Itô-Wiener Chaos Expansion with Exact Residual and Correlation, Variance Inequalities , 1997 .

[11] Inequalities of correlation type for symmetric stable random vectors , 1995, math/9503212.

[12] E. Gluskin. EXTREMAL PROPERTIES OF ORTHOGONAL PARALLELEPIPEDS AND THEIR APPLICATIONS TO THE GEOMETRY OF BANACH SPACES , 1989 .

[13] Stanislaw J. Szarek,et al. Confidence regions for means of multivariate normal distributions and a non-symmetric correlation inequality for gaussian measure , 1997 .