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We find the asymptotic spectral distribution of random Kummer matrix. Then weformulate and prove a~free analogue of HV independence property, which is knownfor classical Kummer and Gamma random variables and for Kummer and Wishartmatrices. We also prove a related characterization of free--Kummer andfree--Poisson (Marchenko-Pastur) non--commutative random variables.

Regression conditions that characterizes free--Poisson and free--Kummer distributions