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

## TL;DR

This paper characterizes free--Poisson and free--Kummer distributions through spectral analysis and independence properties, extending classical results to non-commutative probability and random matrix theory.

## Contribution

It introduces a free analogue of HV independence and provides new characterizations of free--Kummer and free--Poisson distributions.

## Key findings

- Spectral distribution of random Kummer matrix derived
- Free HV independence property formulated and proved
- Characterization of free--Kummer and free--Poisson variables established

## Abstract

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