On convergence of moment generating functions |
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Authors: | NG Ushakov VG Ushakov |
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Institution: | a Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491, Trondheim, Norwayb Department of Mathematical Statistics, Moscow State University, 119992, Moscow, Russia |
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Abstract: | Mukherjea et al. Mukherjea, A., Rao, M., Suen, S., 2006. A note on moment generating functions. Statist. Probab. Lett. 76, 1185-1189] proved that if a sequence of moment generating functions Mn(t) converges pointwise to a moment generating function M(t) for all t in some open interval of the real line, not necessarily containing the origin, then the distribution functions Fn (corresponding to Mn) converge weakly to the distribution function F (corresponding to M). In this note, we improve this result and obtain conditions of the convergence which seem to be sharp: Fn converge weakly to F if Mn(tk) converge to M(tk), k=1,2,…, for some sequence {t1,t2,…} having the minimal and the maximal points. A similar result holds for characteristic functions. |
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Keywords: | Moment generating function Weak convergence |
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