Characterization of the multivariate normal distribution by conditional normal distributions

Kurzfassung/Abstract

Let $X$ and $Y$ be two random vectors with values in $\bbfR\sp k$ and $\bbfR\sp \ell$, respectively. If $Z=(X\sp T,Y\sp T)\sp T$ is multivariate normal, then $X$ given $Y=y$ and $Y$ given $X=x$ are (multivariate) normal; the converse is wrong. Simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vector $Z=(X\sb 1\sp T,\dots,X\sb t\sp T)\sp T$ is splitted into $t\ge 3$ parts $X\sb 1,\dots,X\sb t$.