# Boolean Cumulants and Subordination in Free Probability

## TL;DR

This paper explores subordination in free probability, providing formulas for conditional expectations and explicit distribution calculations for certain free convolutions using Boolean cumulants.

## Contribution

It introduces a new subordination formula involving Boolean cumulants, enabling explicit distribution calculations for free convolutions with functional transformations.

## Key findings

- Derived a formula for conditional expectations as resolvents.
- Enabled explicit calculation of distributions for transformed free sums.
- Generalized subordination formulas for free additive and multiplicative convolutions.

## Abstract

We study subordination of free convolutions. We prove that for free random variables $X,Y$ and a Borel function $f$ the conditional expectation $E_\varphi\left[ (z-X-f(X)Yf^*(X))^{-1}| X\right]$, is a resolvent again. This result allows explicit calculation of the distribution of $X+f(X)Yf^*(X)$. The main tool is a formula for conditional expectations in terms of Boolean cumulant transforms, generalizing subordination formulas for free additive and multiplicative convolutions.