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The Early History of the Cumulants and the Gram‐Charlier Series

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  • Anders Hald

Abstract

The early history of the Gram‐Charlier series is discussed from three points of view: (1) a generalization of Laplace's central limit theorem, (2) a least squares approximation to a continuous function by means of Chebyshev‐Hermite polynomials, (3) a generalization of Gauss's normal distribution to a system of skew distributions. Thiele defined the cumulants in terms of the moments, first by a recursion formula and later by an expansion of the logarithm of the moment generating function. He devised a differential operator which adjusts any cumulant to a desired value. His little known 1899 paper in Danish on the properties of the cumulants is translated into English in the Appendix.

Suggested Citation

  • Anders Hald, 2000. "The Early History of the Cumulants and the Gram‐Charlier Series," International Statistical Review, International Statistical Institute, vol. 68(2), pages 137-153, August.
    • Handle: RePEc:bla:istatr:v:68:y:2000:i:2:p:137-153
    DOI: 10.1111/j.1751-5823.2000.tb00318.x
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    Cited by:

    1. Jiménez, Inés & Mora-Valencia, Andrés & Perote, Javier, 2022. "Semi-nonparametric risk assessment with cryptocurrencies," Research in International Business and Finance, Elsevier, vol. 59(C).
    2. Monica Billio & Bertrand Maillet & Loriana Pelizzon, 2022. "A meta-measure of performance related to both investors and investments characteristics," Annals of Operations Research, Springer, vol. 313(2), pages 1405-1447, June.
    3. Inés Jiménez & Andrés Mora-Valencia & Trino-Manuel Ñíguez & Javier Perote, 2020. "Portfolio Risk Assessment under Dynamic (Equi)Correlation and Semi-Nonparametric Estimation: An Application to Cryptocurrencies," Mathematics, MDPI, vol. 8(12), pages 1-24, November.
    4. Asmussen, Søren & Bladt, Mogens, 2022. "Moments and polynomial expansions in discrete matrix-analytic models," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1165-1188.

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