A note on moment generating functions

In this note, we show that if a sequence of moment generating functions Mn(t)Mn(t) converges pointwise to a moment generating function M(t)M(t) for all t in some open interval of R  , not necessarily containing the origin, then the distribution functions FnFn (corresponding to MnMn) converge weakly to the distribution function F (corresponding to M). The proof uses the basic classical result of Curtiss [1942. A note on the theory of moment generating functions. Ann. Math. Statist. 13 (4), 430–433].