Journal of Mathematical Chemistry

, Volume 56, Issue 6, pp 1631–1655 | Cite as

Multivariate generalized Gram–Charlier series in vector notations

  • Bhaveshkumar C. Dharmani
Original Paper


This article derives the generalized Gram–Charlier (GGC) series in multivariate that expands an unknown joint probability density function (pdf) of a random vector in terms of the differentiations of joint pdf of a known reference random vector. Conventionally, the higher order differentiations of a multivariate pdf and corresponding to it the multivariate GGC series use multi-element array or tensor representations. Instead, the current article derives them in vector notations. The required higher order differentiations of a multivariate pdf are achieved in vector notations through application of a specific Kronecker product based differentiation operator. The resultant multivariate GGC series expression is more compact and more elementary compare to the coordinatewise tensor notations as using vector notations. It is also more comprehensive as apparently more nearer to its counterpart for univariate. Same notations and advantages are shared by other expressions obtained in the article, such as the mutual relations between cumulants and moments of a random vector, integral form of a multivariate pdf, integral form of the multivariate Hermite polynomials, the multivariate Gram–Charlier A series and others. Overall, the article uses only elementary calculus of several variables instead of tensor calculus to achieve the extension of a specific derivation for the GGC series in univariate (Berberan-Santos in J Math Chem 42(3):585–594, 2007) to multivariate.


Multivariate generalized Gram–Charlier (GGC) series Multivariate Gram–Charlier A (GCA) series Multivariate vector Hermite polynomials Kronecker product Vector moments Vector cumulants Charecteristic function 

Mathematics Subject Classification

62E17 approximations to distributions (nonasymptotic) 62H10 distribution of statistics (multivariate analysis) 60E10 characteristic functions; other transforms 62E20 asymptotic distribution theory 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dhirubhai Ambani Institute of Information and Communication TechnologyGandhinagarIndia

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