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On convergence of moment generating functions

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  • Ushakov, N.G.
  • Ushakov, V.G.

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.

Suggested Citation

  • Ushakov, N.G. & Ushakov, V.G., 2011. "On convergence of moment generating functions," Statistics & Probability Letters, Elsevier, vol. 81(4), pages 502-505, April.
    • Handle: RePEc:eee:stapro:v:81:y:2011:i:4:p:502-505
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    References listed on IDEAS

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    1. Mukherjea, A. & Rao, M. & Suen, S., 2006. "A note on moment generating functions," Statistics & Probability Letters, Elsevier, vol. 76(11), pages 1185-1189, June.
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