Convolution, subordination and characterization problems in noncommutative probability

TL;DR

This paper advances the understanding of free probability by generalizing characterization problems through subordination techniques, including new results for distributions of free variables and regressions for monotonically independent variables.

Contribution

It introduces novel subordination methods to generalize known free probability characterizations and provides new distribution characterizations for free and monotonically independent variables.

Findings

Generalized free probability characterizations to unbounded support

Proved a new characterization of free random variable distributions

Analyzed Laha-Lukacs regressions for monotonically independent variables

Abstract

Characterization problems in free probability are studied here. Using subordination of free additive and free multiplicative convolutions we generalize some known characterizations in free probability to random variables with unbounded support. Using this technique we also prove a new characterization of distributions of free random variables. A similar technique is used to study Laha-Lukacs regressions for monotonically independent random variables.