Conditional characterizations of multivariate distributions

Abstract

Let (X, Y) be a bivariate random variable with$X\mathop = \limits^d Y$. Denote the family of conditional distributions ofX givenY by φ(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$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot\cdot}$}}{X} _n = (X_1 ,...,X_n )$ has a multivariate normal distribution, if and only ifX ₁,X ₂ belong to a location-scale family and fori=2, 3, …,n the conditional distribution ofX _i given$X_{i - 1} = x_{i - 1} ,...,X_1 = x_1 is N\left( {\beta _i + \sum\limits_{j = 1}^{i - 1} {\alpha _{j,i - 1} x_j ,\sigma _i^2 } } \right)$, for all realx ₁,…,x _i-1. Also, we show that a conjecture of M. Ahsanullah concerning multivariate normality ofX _n is incorrect.