Berry-Esseen bounds for the multivariate $\mathcal {B}$-free CLT and operator-valued matrices
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- by Marwa Banna and Tobias Mai;
- Trans. Amer. Math. Soc. 376 (2023), 3761-3818
- DOI: https://doi.org/10.1090/tran/8717
- Published electronically: February 3, 2023
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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.References
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Bibliographic Information
- Marwa Banna
- Affiliation: Division of Science, Mathematics, New York University Abu Dhabi, Abu Dhabi, UAE
- MR Author ID: 1103290
- Email: marwa.banna@nyu.edu
- Tobias Mai
- Affiliation: Department of Mathematics, Saarland University, D-66123 Saarbrücken, Germany
- MR Author ID: 984784
- ORCID: 0000-0002-6395-9652
- Email: mai@math.uni-sb.de
- Received by editor(s): May 17, 2021
- Received by editor(s) in revised form: December 31, 2021, and March 11, 2022
- Published electronically: February 3, 2023
- Additional Notes: M. Banna is the corresponding author
This work had been partially supported by the ERC Advanced Grant NCDFP 339760 held by Roland Speicher. - © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 3761-3818
- MSC (2020): Primary 46L54, 60B10, 46L53, 60B20
- DOI: https://doi.org/10.1090/tran/8717
- MathSciNet review: 4586797