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An Independence Property for the Product of GIG and Gamma Laws
Gerard Letac and Jacek Wesolowski
The Annals of Probability
Vol. 28, No. 3 (Jul., 2000), pp. 1371-1383
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2652992
Page Count: 13
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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).
The Annals of Probability © 2000 Institute of Mathematical Statistics