On some properties of normal random variables (Q1073445)

This paper shows that the characterization of bivariate normal distribution given by the authors earlier [Am. Math. Mon. 82, 913-915 (1975; Zbl 0351.60019)] cannot directly be extended to the multivariate case. A generalization is provided in the following theorem. The vector \({\mathfrak X}=(X_ 1,X_ 2,...,X_ n)\) has a multivariate normal distribution if and only if there is a countably infinite set \(A\subseteq {\mathbb{R}}^ n\) such that (i) \({\mathfrak aX}^ T\) is normal for every \({\mathfrak a}\in A\), and (ii) A is dense in an n-dimensional cone. Various examples are given to illustrate the situation in the multivariate case.