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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2411.16103 |
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| _version_ | 1866916502313107456 |
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| author | Díaz, Mario Jaramillo, Arturo |
| author_facet | Díaz, Mario Jaramillo, Arturo |
| contents | We present a straightforward formulation of Stein's method for the semicircular distribution, specifically designed for the analysis of non-commutative random variables. Our approach employs a non-commutative version of Stein's heuristic, interpolating between the target and approximating distributions via the free Ornstein-Uhlenbeck semigroup. A key application of this work is to provide a new perspective for obtaining precise estimates of accuracy in the semicircular approximation for sums of weakly dependent variables, measured under the total variation metric. We leverage the simplicity of our arguments to achieve robust convergence results, including: (i) A Berry-Esseen theorem under the total variation distance and (ii) Enhancements in rates of decay under the non-commutative Wasserstein distance towards the semicircular distribution, given adequate high-order moment matching conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-commutative Stein's Method: Applications to Free Probability and Sums of Non-commutative Variables Díaz, Mario Jaramillo, Arturo Probability 60F05, 46L53, 60G50, 60B10 We present a straightforward formulation of Stein's method for the semicircular distribution, specifically designed for the analysis of non-commutative random variables. Our approach employs a non-commutative version of Stein's heuristic, interpolating between the target and approximating distributions via the free Ornstein-Uhlenbeck semigroup. A key application of this work is to provide a new perspective for obtaining precise estimates of accuracy in the semicircular approximation for sums of weakly dependent variables, measured under the total variation metric. We leverage the simplicity of our arguments to achieve robust convergence results, including: (i) A Berry-Esseen theorem under the total variation distance and (ii) Enhancements in rates of decay under the non-commutative Wasserstein distance towards the semicircular distribution, given adequate high-order moment matching conditions. |
| title | Non-commutative Stein's Method: Applications to Free Probability and Sums of Non-commutative Variables |
| topic | Probability 60F05, 46L53, 60G50, 60B10 |
| url | https://arxiv.org/abs/2411.16103 |