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Mathematical Notes

, Volume 90, Issue 5–6, pp 920–924 | Cite as

A generalization of the Curtiss theorem for moment generating functions

  • A. L. Yakymiv
Article

Abstract

The Curtiss theorem deals with the relation between the weak convergence of probability measures on the line and the convergence of theirmoment generating functions in a neighborhood of zero. We present a multidimensional generalization of this result. To this end, we consider arbitrary σ-finite measures whose moment generating functions exist in a domain of multidimensional Euclidean space not necessarily containing zero. We also prove the corresponding converse statement.

Keywords

probability measure moment generating function Curtiss theorem σ-finite measure analytic function Radon-Nykodym derivative 

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References

  1. 1.
    J. H. Curtiss, “A note on the theory of moment generating functions,” Ann. Math. Statist. 13(4), 430–433 (1942).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    V. N. Sachkov, Probability Methods in Combinatorial Analysis (Nauka, Moscow, 1978) [in Russian].zbMATHGoogle Scholar
  3. 3.
    V. N. Sachkov, Introduction to Combinatorial Methods of Discrete Mathematics (MTsNMO, Moscow, 2004) [in Russian].Google Scholar
  4. 4.
    A. L. Yakymiv, Probabilistic Applications of Tauberian theorems (Fizmatlit, Moscow, 2005) [in Russian].zbMATHGoogle Scholar
  5. 5.
    A. Mukherjea, M. Rao, and S. Suen, “A note on moment generating functions,” Statist. Probab. Lett. 76(11), 1185–1189 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    W. Feller, An Introduction to Probability Theory and Its Applications (J. Wiley, New York, 1971; Mir, Moscow, 1983), Vol. 2.zbMATHGoogle Scholar
  7. 7.
    A. L. Yakymiv, “A limit theorem for the logarithm of the order of a random A-permutation,” Diskretn. Mat. 22(1), 126–149 (2010) [Discrete Math. Appl. 20 (3), 247–275 (2010)].Google Scholar
  8. 8.
    V.-S. Varadarajan, “Measures on topological spaces,” Mat. Sb. 55(1), 35–100 (1961) [Am. Math. Soc. Transl., II. Ser. 48, 161–228 (1965)].MathSciNetGoogle Scholar
  9. 9.
    V. S. Vladimirov, Methods in the Theory of Functions of Several Complex Variables (Nauka, Moscow, 1964) [in Russian].Google Scholar
  10. 10.
    B. V. Shabat, Introduction to Complex Analysis, Pt. 2: Functions of Several Variables (Nauka, Moscow, 1976) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesSteklovRussia

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