Dual Lukacs regressions of negative orders for noncommutative variables

Abstract

In the paper we study characterizations of probability measures in free probability. By constancy of regressions for random variable 𝕍1/2(𝕀 - 𝕌)𝕍1/2 given by 𝕍1/2𝕌𝕍1/2, 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|UV), for i ∈ {-2, -1, 1, 2}. This paper together with previous results18 exhaust all cases of characterizations from Ref. 3.