## An independence property for the product of GIG and gamma laws

### Gérard Letac , Jacek Wesołowski

#### Abstract

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)\\textasciicircum\\-1\\$ and \$V = X\\textasciicircum\\-1\ - (X +Y)\\textasciicircum\\-1\\$ are independent if and only if there exists \$p, a, b \\textgreater\ 0\$ such that \$Y\$ is gamma distributed with shape parameter \$p\$ and scale parameter \$2 a\\textasciicircum\-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)\\textasciicircum\-1\$ and \$V =X\\textasciicircum\-1 -(X +Y)\\textasciicircum\ -1\$ are independent if and only if there exists \$p\\textgreater\(r-1)/2\$ and \$a\$ and \$b\$ in \$V\_+\$ such that \$Y\$ is Wishart distributed with shape parameter \$p\$ and scale parameter \$2a\\textasciicircum\-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).
 Author Gérard Letac Gérard Letac,, - , Jacek Wesołowski (FMIS / DPMS) Jacek Wesołowski,, - Department of Probability and Mathematical Statistics Journal series Annals of Probability, ISSN 0091-1798 Issue year 2000 Vol 28 No 3 Pages 1371-1383 ASJC Classification 1804 Statistics, Probability and Uncertainty; 2613 Statistics and Probability DOI DOI:10.1214/aop/1019160339 URL http://projecteuclid.org/euclid.aop/1019160339 Score (nominal) 30 Publication indicators WoS Citations = 31; Scopus SNIP (Source Normalised Impact per Paper): 2000 = 1.906; WoS Impact Factor: 2006 = 1.301 (2) - 2007=1.349 (5) Citation count* 36 (2015-04-09)
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