Boolean cumulants and subordination in free probability

Abstract

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation [Formula: see text] for free random variables [Formula: see text] and a Borel function [Formula: see text] is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form [Formula: see text]. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

Conditional expectations through Boolean cumulants and subordination – towards a better understanding of the Lukacs property in free probability

“…This section is devoted to calculation of the conditional expectation announced in the introduction. We are able to calculate it using Boolean cumulants and results from [10,7], surprisingly calculations which use Boolean cumulants seems to be simpler than the one which use free cumulants. First we will recall relevant facts about subordination functions, for details see [2].…”

“…Recently discovered connections between Boolean cumulants [7,10] and free probability allows to overcome that difficulty. In particular it was observed that Boolean cumulants appear quite naturally in calculations of conditional expectations of some functions of free random variables.…”

“…In particular it was observed that Boolean cumulants appear quite naturally in calculations of conditional expectations of some functions of free random variables. And we apply the results form [10] to the conditional expectation we are interested in.…”

The Matsumoto-Yor property in free probability via subordination and Boolean cumulants

“…We will need two formulas involving Boolean cumulants. They can be found in [10] and [12] and were used also in [18]. Proposition 2.6.…”

“…Powerful as it is, subordination itself does not allow to prove all the results which we are studying here. We take advantage of connections between free probability and Boolean cumulants established recently in [10,12]. We develop ideas from [12], in particular we provide a new expansion of the reciprocal of the additive subordination function in terms of Boolean cumulants…”