Boolean cumulants and subordination in free probability

Franz Lehner, Kamil Szpojankowski*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 φ (z - X - f(X)Y f (X))-1|X for free random variables X,Y and a Borel function f is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form X + f(X)Y f (X). 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.

Original languageEnglish
Article number21500362
JournalRandom Matrices: Theory and Applications
Issue number4
Early online date1 Jan 2020
Publication statusPublished - 2021


  • Boolean cumulants
  • conditional expectation
  • Free probability
  • subordination

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Discrete Mathematics and Combinatorics


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