The purpose of this paper is to give a few theorems concerning the reciprocal relation between the convergence of a sequence of distribution functions and the convergence of the corresponding sequence of their moment generating functions. The paper consists of two parts. In the first part the univariate case is discussed. The content of this part is closely related to that of a recent paper by J. H. Curtiss [1, p. 430-433], but the results are of a somewhat more general nature, and the methods of proofs are different and do not make use of the theory of a complex variable. The second part deals with the multivariate case which, as far as the author knows, has not been treated before with proofs in as complete and rigorous a way. In both the univariate and multivariate cases the proofs are based on the well known Helly selection principle [2, p. 26] for bounded sequences of monotonic functions.
"On the Convergence of Sequences of Moment Generating Functions." Ann. Math. Statist. 18 (1) 61 - 69, March, 1947. https://doi.org/10.1214/aoms/1177730492