Theorem, Marchenko-Pastur distribution appears in the limit of free version of Poisson limit theorem and is often called free Poisson distribution.

While BP bijection proved to be a powerful tool, it does not preserve all good properties of distributions. Consider for example Lukacs theorem which says that for classically independent random variables $X,Y$ random variables $X+Y$ and $X/(X+Y)$ are independent if and only if $X,Y$ have gamma distribution with the same scale parameter [@Luk55]. One can consider similar problem in free probability and gets the following result (see [@Szp15; @Szp16]) for free random variables $X,Y$ random variables $X+Y$ and $(X+Y)^{-1/2}X(X+Y)^{-1/2}$ are free if and only if $X,Y$ have Marchenko-Pastur (free Poisson) distribution with the same rate. From this example one can see our point - it is not the image under BP bijection of the Gamma distribution (studied in [@PAS08; @HT14]), which has the Lukacs independence property in free probability, but in this context free Poisson distribution plays the role of the classical Gamma distribution.

In [@Szp17] another free independence property was studied – a free version of so called Matsumoto-Yor property (see [@MY01; @LW00]). In classical probability this property says that for independent $X,Y$ random variables $1/(X+Y)$ and $1/X-1/(X+Y)$ 