Dual Lukacs regressions of negative orders for non-commutative variables

TL;DR

This paper characterizes free Poisson and free binomial distributions using regression properties of non-commutative variables, extending classical probability results into free probability theory.

Contribution

It provides the first free probability analogs of classical gamma and beta distribution characterizations via regression constancy, completing the set of known cases.

Findings

Characterization of free Poisson distribution

Characterization of free binomial distribution

Extension of classical regression characterizations to free probability

Abstract

In the paper we study characterizations of probability measures in free probability. By constancy of regressions for random variable $\V(\I-\U)\V$ given by $\V\U\V$, where $\U$ and $\V$ are free, we characterize free Poisson and free binomial distributions. Our paper is analog in free probability of results known in classical probability \cite{BobWes2002Dual}, where gamma and beta distributions are characterized by constancy of $\E((V(1-U))^{i}|UV)$, for $i \in \{-2,-1,1,2\}$. This paper together with previous results \cite{SzpojanWesol} exhaust all cases of characterizations from \cite{BobWes2002Dual}.