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Multivariate Lukacs theorem

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  • Bobecka, Konstancja
  • Wesolowski, Jacek

Abstract

According to the celebrated Lukacs theorem, independence of quotient and sum of two independent positive random variables characterizes the gamma distribution. Rather unexpectedly, it appears that in the multivariate setting, the analogous independence condition does not characterize the multivariate gamma distribution in general, but is far more restrictive: it implies that the respective random vectors have independent or linearly dependent components. Our basic tool is a solution of a related functional equation of a quite general nature. As a side effect the form of the multivariate distribution with univariate Pareto conditionals is derived.

Suggested Citation

  • Bobecka, Konstancja & Wesolowski, Jacek, 2004. "Multivariate Lukacs theorem," Journal of Multivariate Analysis, Elsevier, vol. 91(2), pages 143-160, November.
    • Handle: RePEc:eee:jmvana:v:91:y:2004:i:2:p:143-160
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    References listed on IDEAS

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    1. Griffiths, R. C., 1984. "Characterization of infinitely divisible multivariate gamma distributions," Journal of Multivariate Analysis, Elsevier, vol. 15(1), pages 13-20, August.
    2. Arnold, Barry C., 1987. "Bivariate distributions with pareto conditionals," Statistics & Probability Letters, Elsevier, vol. 5(4), pages 263-266, June.
    3. Shun-Hwa Li & Wen-Jang Huang & Mong-Na Huang, 1994. "Characterizations of the Poisson process as a renewal process via two conditional moments," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(2), pages 351-360, June.
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    Cited by:

    1. Ferrari, A. & Letac, G. & Tourneret, J.-Y., 2007. "Exponential families of mixed Poisson distributions," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1283-1292, July.

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