Conditions are given for the strong consistency and asymptotic normality of the MLE (maximum likelihood estimator) for multiparameter exponential models. Because of the special structure assumed, the conditions are less restrictive than required by general theorems in this area. The technique involves certain convex functions on Euclidean spaces that arise naturally in the present context. Some examples are considered; among them, the multinomial distribution. Some convexity and continuity properties of multivariate cumulant generating functions are also discussed.
"Consistency and Asymptotic Normality of MLE's for Exponential Models." Ann. Math. Statist. 43 (1) 193 - 204, February, 1972. https://doi.org/10.1214/aoms/1177692713