CHARACTERIZATIONS OF GAUSSIAN DISTRIBUTIONS ON SIMPLY CONNECTED NILPOTENT LIE-GROUPS AND SYMMETRICAL SPACES

For simply connected nilpotent Lie groups, we show that a probability measure is gaussian in the sense of Bernstein (for a definition thereo f which in a natural way involves non-commutativity) iff it is a gauss ian measure in the classical sense concentrated on an abelian subgroup . Furthermore we carry over the Skitovic-Darmois theorem to symmetric spaces of non-compact type.