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 ϕ " pz ´X ´f pXqY f ˚pX qq ´1|X ‰ , is a resolvent again. This result allows explicit calculation of the distribution of X`f pXqY f ˚pX q. The main tool is a formula for conditional expectations in terms of Boolean cumulant transforms, generalizing subordination formulas for free additive and multiplicative convolutions.