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[ (z-X-f(X)Yf*(X))-1| X], 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.
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