Conditional characterizations of multivariate distributions

Let (X, Y) be a bivariate random variable with {Mathematical expression}. Denote the family of conditional distributions of X given Y by (x|y) determines the joint distribution of (X, Y) in such a setting. This observation provides an alternative proof of a bivariate normal characterization first proved by Ahsanullah (1985). Many analogous characterization results can be enumerated. Some examples are provided. Multivariate extensions are discussed. For example it is shown that a random vector {Mathematical expression} has a multivariate normal distribution, if and only if X 1, X 2 belong to a location-scale family and for i=2, 3, ..., n the conditional distribution of X i given {Mathematical expression}, for all real x 1,..., x i-1. Also, we show that a conjecture of M. Ahsanullah concerning multivariate normality of X n is incorrect. 1988 Physica-Verlag Ges.m.b.H.

Conditional characterizations of multivariate distributions Academic Article