On the Lukacs property for free random variables

Abstract

Abstract. The Lukacs property of the free Poisson distribution is studied here. We prove that if free X and Y are free Poisson distributed with suitable parameters, then X + Y and (X + Y) − 1 2 X (X + Y) − 1 2 are free. As as an auxiliary result we give joint cumulants of X and X −1 for free Poisson distributed X.

Conditional expectations through Boolean cumulants and subordination – towards a better understanding of the Lukacs property in free probability

“…The proof presented below uses a result from [20]. The original proof of the Lukacs property for free Poisson distribution from [19] relies heavily on combinatorics of free probability. In particular we calculated there joint free cumulants of X and X −1 for free Poisson distributed X.…”

“…The analogue of Bernstein's theorem was studied by Nica (see [13]) and says that for free X, Y , random variables U = X + Y and V = X − Y are free if and only if X, Y have Wigner semicircle distributions with the same variance. The free analogue of Lukacs theorem was studied in [19].…”

“…That is we show that for free random variables X and Y both having free Poisson (Marchenko-Pastur) distribution, random variables X + Y and (X + Y ) −1/2 X(X + Y ) −1/2 are free. In [19] we presented a "hands on", direct combinatorial proof, our strategy was to show vanishing of mixed free cumulants of X + Y and (X + Y ) −1/2 X(X + Y ) −1/2 . In particular we calculated explicitly joint free cumulants of X, X −1 for invertible, free Poisson distributed random variable X.…”

Remarks on a Free Analogue of the Beta Prime Distribution

“…Another free analogue of characterization problem by independence, in [34], [35], that is, a free version of Lukacs theorem, the classical version states that, for independent random variables X and Y , the random variables X + Y and X X + Y are independent if and only if X and Y are gamma distributed with the same scale parameter [25]. Here one should note that in the free analogue of Lukacs theorem, the free Poisson distribution plays the classical gamma distribution role.…”

A Matsumoto–Yor characterization for Kummer and Wishart random matrices

“…There are other independence characterizations involving Kummer distribution, which use different methods [26,21]; see also a characterization of vector-variate Kummer law in [20]. Finally, it is important to note that in some sense one can pass with the rank r of S + to infinity and obtain properties and characterizations of laws of free random variables; see [22] for Lukacs and [23] for Matsumoto-Yor property in free probability.…”