## 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.

### MSC:

 62E10 Characterization and structure theory of statistical distributions 62H05 Characterization and structure theory for multivariate probability distributions; copulas 60E05 Probability distributions: general theory

Zbl 0983.60075
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### References:

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