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Multivariate quadratic forms of random vectors

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  • Blacher, René

Abstract

We obtain the distribution of the sum of n random vectors and the distribution of their quadratic forms: their densities are expanded in series of Hermite and Laguerre polynomials. We do not suppose that these vectors are independent. In particular, we apply these results to multivariate quadratic forms of Gaussian vectors. We obtain also their densities expanded in Mac Laurin series or in the form of an integral. By this last result, we introduce a new method of computation which can be much simpler than the previously known techniques. In particular, we introduce a new method in the very classical univariate case. We remark that we do not assume the independence of normal variables.

Suggested Citation

  • Blacher, René, 2003. "Multivariate quadratic forms of random vectors," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 2-23, October.
    • Handle: RePEc:eee:jmvana:v:87:y:2003:i:1:p:2-23
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    References listed on IDEAS

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    1. Mathai, A. M. & Moschopoulos, P. G., 1991. "On a multivariate gamma," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 135-153, October.
    2. Coelho, Carlos A., 1998. "The Generalized Integer Gamma Distribution--A Basis for Distributions in Multivariate Statistics," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 86-102, January.
    3. Royen, T., 1991. "Expansions for the multivariate chi-square distribution," Journal of Multivariate Analysis, Elsevier, vol. 38(2), pages 213-232, August.
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    Cited by:

    1. Matthieu Garcin & Dominique Guegan, 2013. "Probability density of the wavelet coefficients of a noisy chaos," Post-Print hal-00800997, HAL.
    2. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    3. Matthieu Garcin & Dominique Guegan, 2013. "Probability density of the wavelet coefficients of a noisy chaos," Documents de travail du Centre d'Economie de la Sorbonne 13015, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    4. Song, Iickho & Lee, Seungwon, 2015. "Explicit formulae for product moments of multivariate Gaussian random variables," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 27-34.
    5. Regoli, Giuliana, 2009. "A class of bivariate exponential distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1261-1269, July.

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