Three dual regression schemes for the Lukacs theorem

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

“…It was also observed (see [9,3,22]) that the strong assumption that both vectors (X, Y ) and (U, V ) consist of independent (respectively free) random variables can be weakened and it is enough to assume only constancy of some conditional moments of U given V . This phenomenon was observed both in context of commutative, independent random variables, as well as for non-commutative, free random variables (c.f.…”

Convolution, subordination and characterization problems in noncommutative probability

“…Next theorem gives a similar generalization of a result from [19] where authors proved characterization of free binomial and free Piosson random variable. This result is a free analogue of classical probability characterization proved in [4]. Theorem 3.4.…”

Dual Lukacs regressions of negative orders for noncommutative variables

“…By constancy of regressions for random variable V 1/2 (I − U)V 1/2 given by V 1/2 UV 1/2 , where U and V are free, we characterize free Poisson and free binomial distributions. Our paper is a free probability analogue of results known in classical probability [3], where gamma and beta distributions are characterized by constancy of E (V (1 − U )) i |U V , for i ∈ {−2, −1, 1, 2}. This paper together with previous results [18] exhaust all cases of characterizations from [3].…”

“…The technique used in the proofs develops method from [18]. These two theorems below complete free probability analogues of classical results from [3].…”

“…In the previous paper [18] we investigated a free analogue of dual Lukacs regressions from [3] in the case (i, j) = (1, 2). Our aim in this paper is to prove free analogues for the remaining cases.…”