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On the density of the sum of two independent Student t-random vectors

Author

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  • Berg, C.
  • Vignat, C.

Abstract

In this paper, we find an expression for the density of the sum of two independent d-dimensional Student t-random vectors and with arbitrary degrees of freedom. As a byproduct we also obtain an expression for the density of the sum , where is normal and is an independent Student t-vector. In both cases the density is given as an infinite series where fn is a sequence of probability densities on and (cn) is a sequence of positive numbers of sum 1, i.e. the distribution of a non-negative integer-valued random variable C, which turns out to be infinitely divisible for d=1 and d=2. When d=1 and the degrees of freedom of the Student variables are equal, we recover an old result of Ruben.

Suggested Citation

  • Berg, C. & Vignat, C., 2010. "On the density of the sum of two independent Student t-random vectors," Statistics & Probability Letters, Elsevier, vol. 80(13-14), pages 1043-1055, July.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:13-14:p:1043-1055
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    References listed on IDEAS

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    1. Witkovský, Viktor, 2002. "Exact distribution of positive linear combinations of inverted chi-square random variables with odd degrees of freedom," Statistics & Probability Letters, Elsevier, vol. 56(1), pages 45-50, January.
    2. Nicola Cufaro Petroni, 2007. "Mixtures in non stable Levy processes," Papers math/0702058, arXiv.org.
    3. Nason, Guy P., 2006. "On the sum of t and Gaussian random variables," Statistics & Probability Letters, Elsevier, vol. 76(12), pages 1280-1286, July.
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    Cited by:

    1. Simos G. Meintanis, 2020. "Comments on: Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 898-902, December.
    2. Tatsuya Kubokawa & Éric Marchand & William E. Strawderman, 2014. "On Predictive Density Estimation for Location Families under Integrated L 2 and L 1 Losses," CIRJE F-Series CIRJE-F-935, CIRJE, Faculty of Economics, University of Tokyo.
    3. Kubokawa, Tatsuya & Marchand, Éric & Strawderman, William E., 2015. "On predictive density estimation for location families under integrated squared error loss," Journal of Multivariate Analysis, Elsevier, vol. 142(C), pages 57-74.

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