An independence property for the product of GIG and gamma laws

• 成果类型: Article
• 署名作者: Letac, G; Wesolowski, J
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Warsaw University of Technology; Polish Academy of Sciences; Institute of Mathematics of the Polish Academy of Sciences; University of Warsaw
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 1371-1383
• 关键词: inverse gaussian distribution; wishart distributions; continued fractions; symmetrical cones

摘要

Matsumoto and Yor have recently discovered an interesting transformation which preserves a bivariate probability measure which is a product of the generalized inverse Gaussian (GIG) and gamma distributions. This paper is devoted to a detailed study of this phenomenon. Let X and Y be two independent positive random variables. We prove (Theorem 4.1) that U = (X + Y)(-1) and V = X-1 - (X + Y)(-1) are independent if and only if there exists p, a, b > 0 such that Y is gamma distributed with shape parameter p and scale parameter 2a(-1), and such that X has a GIG distribution with parameters -p, a and b (the direct part for a = b was obtained in Matsumoto and Yor). The result is partially extended (Theorem 5.1) to the case where X and Y are valued in the cone V, of symmetric positive definite (r, r) real matrices as follows: under a hypothesis of smoothness of densities, we prove that U = (X + Y)(-1) and V = X-1 - (:X + Y)(-1) are independent if and only if there exists p > (r - 1)/2 and a and b in V+ such that Y is Wishart distributed with shape parameter p and scale parameter 2a(-1), and such that X has a matrix GIG distribution with parameters -p, a and b. The direct result is;also extended to singular Wishart distributions (Theorem 3.1).