# Monotone, free, and boolean cumulants: a shuffle algebra approach

## TL;DR

This paper explores the combinatorial Hopf algebra framework to unify and analyze monotone, free, and boolean cumulants using shuffle algebra structures and the pre-Lie Magnus expansion.

## Contribution

It introduces a novel shuffle algebra approach to cumulants, connecting different types via the pre-Lie Magnus expansion and half-shuffle logarithms.

## Key findings

- Unified algebraic framework for cumulants
- Explicit relations between cumulant types
- Application of shuffle and half-shuffle exponentials

## Abstract

The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each other using the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.