J Austral Math Soc Ser A 43 pp366--374, 1987.

A Characterization and a Variational Inequality for the Multivariate Normal Distribution

Wolfgang Stadje

Abstract

Various generalizations of the Maxwell characterization of the multivariate standard normal distribution are derived. For example the following is proved: If for a k-dimensional random vector X there exists an n Î {1, ..., k - 1} such that for each n-dimensional linear subspace H Ì R^k the projections of X on H and H^{^} are independent, X is normal. If X has a rotationally symmetric density and its projection on some H has a density of the same functional form, X is normal. Finally we give a variational inequality for the multivariate normal distribution which resembles the isoperimetric inequality for the surface measure on the sphere.

1980 AMS Subject Classification: 62H05

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Authors

Wolfgang Stadje

Fachbereich Mathematik, Universität Osnabrück, Albrechtstrasse 28, West Germany.