An estimate for $β$-Hermite ensembles via the zeros of Hermite polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929706135191552 |
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| author | Voit, Michael |
| author_facet | Voit, Michael |
| contents | Let $X$ be an $N$-dimensional random vector which describes the ordered eigenvalues of a $β$-Hermite ensemble, and let $z$ the vector containing the ordered zeros of the Hermite poynomial $H_N$. We present an explicit estimate for $P(\|X-z\|_2\geε)$ for small $ε>0$ and large parameters $β$. The proof is based on a central limit theorem for these ensembles for $β\to\infty$ with explicit eigenvalues of the covariance matrices of the limit. The estimate is similar to previous estimates of Dette and Imhof (2009). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05886 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An estimate for $β$-Hermite ensembles via the zeros of Hermite polynomials Voit, Michael Probability Mathematical Physics 60B20, 60F05, 82C22, 33C45 Let $X$ be an $N$-dimensional random vector which describes the ordered eigenvalues of a $β$-Hermite ensemble, and let $z$ the vector containing the ordered zeros of the Hermite poynomial $H_N$. We present an explicit estimate for $P(\|X-z\|_2\geε)$ for small $ε>0$ and large parameters $β$. The proof is based on a central limit theorem for these ensembles for $β\to\infty$ with explicit eigenvalues of the covariance matrices of the limit. The estimate is similar to previous estimates of Dette and Imhof (2009). |
| title | An estimate for $β$-Hermite ensembles via the zeros of Hermite polynomials |
| topic | Probability Mathematical Physics 60B20, 60F05, 82C22, 33C45 |
| url | https://arxiv.org/abs/2502.05886 |