Distributions of noncommutative polynomials in correlated sums of freely independent elements with amalgamation

Noncommutative distributions transfer the well-established notion of joint distributions known from classical probability theory to the realm of noncommutative probability. We are mainly interested by polynomials in noncommuting variables which constitute a very natural class of noncommutative functions. Using linearisation techniques, we push the noncommutative Lindeberg method to the operator-valued setting to study polynomials in correlated sums of freely independent elements with amalgamation. The summands do not need to be identically distributed and are also allowed to have different covariances. We give explicit rates of convergence on the associated Cauchy transforms to polynomials in a family of operator-valued semicircular elements.
As applications, we first give a Berry-Esseen type theorem for the speed of convergence in the multivariate free central limit theorem. We also consider polynomials in correlated matrices having free entries and a (co)variance profile. We give some precise cases where the limiting cumulative distribution is Hölder continuous and thus the quantitative bounds on the Cauchy transforms can be passed to the Kolmogorov distance.
(Based on an forthcoming joint work with Tobias Mai)