A Generalization of the Curtiss Theorem for Moment Generating Functions

The Curtiss theorem deals with the relation between the weak convergence of probability measures on the line and the convergence of their moment generating functions in a neighborhood of zero. We present a multidimensional generalization of this result. To this end, we consider arbitrary $\sigma$-finite measures whose moment generating functions exist in a domain of multidimensional Euclidean space not necessarily containing zero. We also prove the corresponding converse statement.