Independence properties of the Matsumoto--Yor type

Details Independence properties of the Matsumoto--Yor type Independence properties of the Matsumoto--Yor type

2012-03-02

arxiv.org/abs/1203.0381v1

Bernoulli 2012, Vol. 18, No. 1, 119-136

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/10-BEJ325 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ325

We define Letac-Wesolowski-Matsumoto-Yor (LWMY) functions as decreasing functions from $(0,\infty)$ onto $(0,\infty)$ with the following property: there exist independent, positive random variables $X$ and $Y$ such that the variables $f(X+Y)$ and $f(X)-f(X+Y)$ are independent. We prove that, under additional assumptions, there are essentially four such functions. The first one is $f(x)=1/x$. In this case, referred to in the literature as the Matsumoto-Yor property, the law of $X$ is generalized inverse Gaussian while $Y$ is gamma distributed. In the three other cases, the associated densities are provided. As a consequence, we obtain a new relation of convolution involving gamma distributions and Kummer distributions of type 2.

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