On convergence of moment generating functions

N.G. Ushakov and V.G. Ushakov

Statistics & Probability Letters, 2011, vol. 81, issue 4, 502-505

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.

Keywords: Moment; generating; function; Weak; convergence (search for similar items in EconPapers)
Date: 2011
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