%0 Generic %1 koudou2012independence %A Koudou, A. E. %A Vallois, P. %D 2012 %K distributions probability statistics %R 10.3150/10-BEJ325 %T Independence properties of the Matsumoto--Yor type %U http://arxiv.org/abs/1203.0381 %X We define Letac-Wesolowski-Matsumoto-Yor (LWMY) functions as decreasing functions from $(0,ınfty)$ onto $(0,ınfty)$ 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.

@misc{koudou2012independence, abstract = {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.}, added-at = {2015-05-21T15:55:02.000+0200}, author = {Koudou, A. E. and Vallois, P.}, biburl = {https://www.bibsonomy.org/bibtex/2b991ac97b531e56fcb3b39a4dee1cbd0/shabbychef}, description = {[1203.0381] Independence properties of the Matsumoto--Yor type}, doi = {10.3150/10-BEJ325}, interhash = {b308ab95a9543e4b382b3e593715d891}, intrahash = {b991ac97b531e56fcb3b39a4dee1cbd0}, keywords = {distributions probability statistics}, note = {cite arxiv:1203.0381Comment: Published in at http://dx.doi.org/10.3150/10-BEJ325 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)}, timestamp = {2015-05-21T15:55:02.000+0200}, title = {Independence properties of the Matsumoto--Yor type}, url = {http://arxiv.org/abs/1203.0381}, year = 2012 }