##### Department of Mathematics,

University of California San Diego

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### Math 243 - Functional Analysis Seminar

## Marwa Banna

#### NYU Abu Dhabi

## Berry-Esseen Bounds for Operator-valued Free Limit Theorems

##### Abstract:

The development of free probability theory has drawn much inspiration from its deep and far reaching analogy with classical probability theory. The same holds for its operator-valued extension, where the fundamental notion of free independence is generalized to free independence with amalgamation as a kind of conditional version of the former. Its development naturally led to operator-valued free analogues of key and fundamental limiting theorems such as the operator-valued free Central Limit Theorem due to Voiculescu and results about the asymptotic behaviour of distributions of matrices with operator-valued entries. In this talk, we show Berry-Esseen bounds for such limit theorems. The estimates are on the level of operator-valued Cauchy transforms and the L{\'e}vy distance. We address also the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the L{\'e}vy distance for the operator-valued free CLT. This also yields quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile. This is a joint work with Tobias Mai.

Host: David Jekel

### November 30, 2021

### 8:00 AM

### Please email djekel@ucsd.edu for Zoom details.

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