Boolean Cumulants and Subordination in Free Probability

Abstract

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

Numerous corrections and improvements suggested by an anonymous referee; 34 pages, 5 figures

References in corpus (4)

• Differential Equations for Algebraic Functions
• Relations between cumulants in noncommutative probability
• Cumulants, free cumulants and half-shuffles
• An Operad of Non-commutative Independences Defined by Trees