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A note on moment generating functions. (English) Zbl 1092.60008
Summary: We show that if a sequence of moment generating functions \(M_{n}(t)\) converges pointwise to a moment generating function \(M(t)\) for all \(t\) in some open interval of \(R\), not necessarily containing the origin, then the distribution functions \(F_{n}\) (corresponding to \(M_{n})\) converge weakly to the distribution function \(F\) (corresponding to \(M\)). The proof uses the basic classical result of J. H. Curtiss [Ann. Math. Stat. 13, No. 4, 430–433 (1942; Zbl 0063.01024)].

MSC:
60E10 Characteristic functions; other transforms
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References:
[1] Cramer, H., 1937. Random Variables and Probability Distributions. Cambridge. · JFM 63.1080.01
[2] Curtiss, J.H., A note on the theory of moment generating functions, Ann. math. statist., 13, 4, 430-433, (1942) · Zbl 0063.01024
[3] Lévy, P., 1937. Théorie de l’Addition des Variables Aléatoires. Paris. · JFM 63.0490.04
[4] Mutafchiev, L.R., Large distinct part sizes in a random integer partition, Acta math. hungar., 87, 1-2, 47-69, (2000) · Zbl 1062.11063
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