Statistics & Probability Letters 19 (1994) 313-315 North-Holland 15 March 1994 A conditional characterization of the multivariate normal distribution Barry C. Arnold zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA University of California, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Riverside, USA Enrique Castillo and Jose Maria Sarabia University of Cantabria, Santander, Spain Received December 1992 Revised April 1993 Abstract: If X is a k-dimensional random vector, we denote by Xcl, j) the vector X with coordinates i and j deleted. If for each i, j the conditional distribution of X,, X, given Xcl, j) = xci, j) is classical bivariate normal for each xci, j) GZ IWke2 then it is shown that X has a classical k-variate normal distribution. Keywords: Bivariate conditionals; classical normal distribution; conditional specification zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ 1. Introduction If X has a classical k-dimensional normal distri- bution with mean vector p and variance covari- ante matrix ,Z then it is well known that all marginals of the distribution of X are again clas- sical multivariate normal, It is also true that all conditional distributions of X given X (where X and X are non-overlapping subvectors of X) are multivariate normal. Efforts to characterize k-di- mensional normality assuming only marginal or conditional normal specifications have been, in general, unsuccessful. If we assume only that X i, . . . , X, are marginally univariate normal then an enormous panoply of k dimensional distribu- tions can be constructed with such a marginal normal specification. If we require that higher dimensional marginals are all multivariate normal we still fail to guarantee k-dimensional normality. Correspondence too: Dr. B.C. Arnold, University of California, Department of Statistics, Riverside, CA 92521-0138, USA. The classic example is provided by the following joint distribution. fx( x) = (ZT) -k/Q ,-cL~5/* x 1+ fixi Z()xJ<l,Vi) . H 1 i=l zyxwvutsrqponmlkjihgfedcbaZYXWV 1 This clearly has all marginals of order <k being normal but it fails to be k-dimensional normal. Conditional specification of the k-dimensional distribution seems to be similarly fraught with difficulties. Following Castillo and Galambos (1989) and in more detail Arnold, Castillo and Sarabia (19921, the specification of conditional normality for each Xi given Xci, does not guaran- tee normality for X. It does determine the nature of the joint density namely zyxwvutsrqponmlkjihgfedcbaZ fAx> 0167-7152/94/$07.00 0 1994 - Elsevier Science B.V. All rights resewed SSDI 0167-7152(93)E0120-I 313