Weighted Sums and Berry-Esseen type estimates in Free Probability Theory
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909293773586432 |
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| author | Neufeld, Leonie |
| author_facet | Neufeld, Leonie |
| contents | We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner's semicircle law is of order $n^{-1/2}$ with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order $n^{-1}$, thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalize to the setting of sums of free non-identically distributed bounded self-adjoint random variables leading to a new rate of convergence in the free central limit theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_06489 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weighted Sums and Berry-Esseen type estimates in Free Probability Theory Neufeld, Leonie Probability Operator Algebras 46L54, 60E05 We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner's semicircle law is of order $n^{-1/2}$ with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order $n^{-1}$, thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalize to the setting of sums of free non-identically distributed bounded self-adjoint random variables leading to a new rate of convergence in the free central limit theorem. |
| title | Weighted Sums and Berry-Esseen type estimates in Free Probability Theory |
| topic | Probability Operator Algebras 46L54, 60E05 |
| url | https://arxiv.org/abs/2303.06489 |