Dual Lukacs regressions for non-commutative variables

Abstract

Dual Lukacs type characterizations of random variables in free probability are studied here. First, we develop a freeness property satisfied by Lukacs type transformations of free-Poisson and free-Binomial non-commutative variables which are free. Second, we give a characterization of non-commutative free-Poisson and free-Binomial variables by properties of first two conditional moments, which mimic Lukacs type assumptions known from classical probability. More precisely, our result is a non-commutative version of the following result known in classical probability: if $U$, $V$ are independent real random variables, such that $E(V(1-U)|UV)$ and $E(V^2(1-U{)}^2|UV)$ are non-random then $V$ has a gamma distribution and $U$ has a beta distribution.

References in corpus (6)

• On a class of free Levy laws related to a regression problem
• Conditional moments of q-Meixner processes
• Markov processes with free Meixner laws
• The Lukacs-Olkin-Rubin theorem on symmetric cones through Gleason's theorem
• Extension of the Olkin and Rubin Characterization to the Wishart distribution on homogeneous cones
• Free Quadratic Harness

Cited by in corpus (6)

• Sample Variance in Free Probability
• Convolution, subordination and characterization problems in noncommutative probability
• A Characterization of the Normal Distribution by the Independence of a Pair of Random Vectors
• Dual Lukacs regressions of negative orders for non-commutative variables
• New characterization of two-state normal distribution
• Remarks on a free analogue of the beta prime distribution