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**An independence property for the product of GIG and gamma laws.**
*(English)*
Zbl 1010.62010

Summary: H. Matsumoto and M. Yor [ Nagoya Math. J. 162, 65-86 (2001; Zbl 0983.60075)] 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 that \(U=(X+Y)^{-1}\) and \(V=X^{-1}-(X+Y)^{-1}\) are independent if and only if there exist \(p,a,b>0\) 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 by Matsumoto and Yor). The result is partially extended to the case where \(X\) and \(Y\) take values in the cone \(V_+\) of symmetric positive definite \(r\times r\) real matrices as follows:

Under the 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 exist \(p>(r-1)/2\) 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.

Let \(X\) and \(Y\) be two independent, positive random variables. We prove that \(U=(X+Y)^{-1}\) and \(V=X^{-1}-(X+Y)^{-1}\) are independent if and only if there exist \(p,a,b>0\) 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 by Matsumoto and Yor). The result is partially extended to the case where \(X\) and \(Y\) take values in the cone \(V_+\) of symmetric positive definite \(r\times r\) real matrices as follows:

Under the 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 exist \(p>(r-1)/2\) 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.

### MSC:

62E10 | Characterization and structure theory of statistical distributions |

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

60E05 | Probability distributions: general theory |

### Citations:

Zbl 0983.60075
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\textit{G. Letac} and \textit{J. Wesołowski}, Ann. Probab. 28, No. 3, 1371--1383 (2000; Zbl 1010.62010)

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### References:

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