## Abstract

We provide bounds of Berry-Esseen type for fundamental limit theorems in operator-valued free probability theory such as the operator-valued free Central Limit Theorem and the asymptotic behaviour of distributions of operator-valued matrices. Our estimates are on the level of operator-valued Cauchy transforms and the Lévy distance. We address the single-variable as well as 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 Lévy distance for the operator-valued free Central Limit Theorem. Our results also yield quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile. In the scalar-valued multivariate case, these estimates could be passed to explicit bounds on the order of convergence under the Kolmogorov distance.

Original language | English (US) |
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Pages (from-to) | 3761-3818 |

Number of pages | 58 |

Journal | Transactions of the American Mathematical Society |

Volume | 376 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2023 |

## Keywords

- Berry-Esseen bounds
- Kolmogorov distance
- Lindeberg method
- Lévy distance
- Noncommutative distributions
- linear matrix pencils
- linearizations
- noncommutative polynomials
- operator-valued matrices
- operator-valued multivariate free CLT
- operator-valued semicircular family

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics