Joint cumulants for natural independence

TL;DR

This paper introduces a unified approach to joint cumulants in non-commutative probability, covering five types of natural independence, and extends existing formulas to multivariate cases.

Contribution

It provides a new unified framework for joint cumulants across various natural independences and extends Muraki's formula to multivariate scenarios.

Findings

Unified treatment of joint cumulants for tensor, free, Boolean, monotone, and anti-monotone independence.

New characterization of joint cumulants for all types of natural independence.

Extension of Muraki's formula to multivariate monotone independent variables.

Abstract

Many kinds of independence have been defined in non-commutative probability theory. Natural independence is an important class of independence; this class consists of five independences (tensor, free, Boolean, monotone and anti-monotone ones). In the present paper, a unified treatment of joint cumulants is introduced for natural independence. The way we define joint cumulants enables us not only to find the monotone joint cumulants but also to give a new characterization of joint cumulants for other kinds of natural independence, i.e., tensor, free and Boolean independences. We also investigate relations between generating functions of moments and monotone cumulants. We find a natural extension of the Muraki formula, which describes the sum of monotone independent random variables, to the multivariate case.