Calculating Cumulants of a Taylor Expansion of a Multivariate Function

Kamanzi‐wa‐Binyavanga

International Statistical Review, 2009, vol. 77, issue 2, 212-221

Abstract: A method, which we believe is simpler and more transparent than the one due to McCullagh (1984), is described for obtaining the cumulants of a scalar multivariate stochastic Taylor expansion. Its generalisation is also suggested. An important feature, previously not reported, is that the expansion of every cumulant of order≥ 2 is made up of separate subseries. In order to handle certain frequently occurring sums over permutations of members of compound index sets, we introduce a new notation [m]*, where m is a positive integer. Une méthode plus simple et plus transparente que celle de McCullagh (1984), est décrite pour obtenir les cumulants d'une expansion de Taylor scalaire, stochastique à plusieurs variables. Sa généralisation est aussi suggérée. Une caractéristique importante pas signalée auparavant est que l'expansion de chaque cumulant de l'ordre ≥ 2 est composée de sous‐séries séparées. Pour pouvoir traiter certaines sommes qui apparaissent fréquemment sur des permutations des membres des ensembles d'indices composés, nous introduisons une nouvelle notation [m], où m est un entier relatif appositif.

Date: 2009
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https://doi.org/10.1111/j.1751-5823.2008.00058.x

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