Title: Independencies for Kummer and gamma distributons
Authors: Jacek Wesolowski - Warsaw University of Technology (Poland) [presenting]
Abstract: While searching for analogs of the Matsumoto-Yor property, i.e. independence of $X+Y$ and $1/X - 1/(X+Y)$ for independent $X$ and $Y$ with generalized inverse Gaussian and gamma distribution, it was previously discovered that for independent $X$ and $Y$ with Kummer and gamma distributions, respectively, random variables $V=X+Y$ and $U=(1+1/(X+Y))/(1+1/X)$ are independent and have Kummer and beta distributions. Similar, but different property was discovered later: if $X$ and $Y$ are independent Kummer and gamma random variables then $V=Y/(1+X)$ and $U=X(1+Y/(1+X))$ are also independent and have Kummer and gamma distributions. We will consider different versions of converse results, some concerned just with independence properties of $U$ and $V$, some with a weaker assumption of constancy of regressions of $U$ given $V$. Also a multivariate version of the second property and a related characterization will be presented. It involves a new multivariate version of the Kummer law and tree-based transformations.