Independence properties of the Matsumoto--Yor type
A. Koudou, and P. Vallois. (2012)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).
DOI: 10.3150/10-BEJ325
Abstract
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.
Description
[1203.0381] Independence properties of the Matsumoto--Yor type
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)
%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
}