Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory
In this paper, we consider the asymptotic behavior of X(n)fn≔∑ni=1fn(xi)Xfn(n)≔∑i=1nfn(xi), where xi,i=1,…,n form orthogonal polynomial ensembles and fn is a real-valued, bounded measurable function. Under the condition that VarX(n)fn→∞VarXfn(n)→∞, the Berry-Esseen (BE) bound and Cramér type moderat...
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| Format: | Journal Article |
| Language: | English |
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2017
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| Online Access: | https://hdl.handle.net/10356/86704 http://hdl.handle.net/10220/44176 |
| _version_ | 1826109454072938496 |
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| author | Pan, Guangming Wang, Shaochen Zhou, Wang |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Pan, Guangming Wang, Shaochen Zhou, Wang |
| author_sort | Pan, Guangming |
| collection | NTU |
| description | In this paper, we consider the asymptotic behavior of X(n)fn≔∑ni=1fn(xi)Xfn(n)≔∑i=1nfn(xi), where xi,i=1,…,n form orthogonal polynomial ensembles and fn is a real-valued, bounded measurable function. Under the condition that VarX(n)fn→∞VarXfn(n)→∞, the Berry-Esseen (BE) bound and Cramér type moderate deviation principle (MDP) for X(n)fnXfn(n) are obtained by using the method of cumulants. As two applications, we establish the BE bound and Cramér type MDP for linear spectrum statistics of Wigner matrix and sample covariance matrix in the complex cases. These results show that in the edge case [which means fn has a particular form f(x)I(x≥θn)f(x)I(x≥θn) where θnθn is close to the right edge of equilibrium measure and f is a smooth function], X(n)fnXfn(n) behaves like the eigenvalues counting function of the corresponding Wigner matrix and sample covariance matrix, respectively. |
| first_indexed | 2024-10-01T02:18:29Z |
| format | Journal Article |
| id | ntu-10356/86704 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T02:18:29Z |
| publishDate | 2017 |
| record_format | dspace |
| spelling | ntu-10356/867042023-02-28T19:23:40Z Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory Pan, Guangming Wang, Shaochen Zhou, Wang School of Physical and Mathematical Sciences Probability Theory Correlation In this paper, we consider the asymptotic behavior of X(n)fn≔∑ni=1fn(xi)Xfn(n)≔∑i=1nfn(xi), where xi,i=1,…,n form orthogonal polynomial ensembles and fn is a real-valued, bounded measurable function. Under the condition that VarX(n)fn→∞VarXfn(n)→∞, the Berry-Esseen (BE) bound and Cramér type moderate deviation principle (MDP) for X(n)fnXfn(n) are obtained by using the method of cumulants. As two applications, we establish the BE bound and Cramér type MDP for linear spectrum statistics of Wigner matrix and sample covariance matrix in the complex cases. These results show that in the edge case [which means fn has a particular form f(x)I(x≥θn)f(x)I(x≥θn) where θnθn is close to the right edge of equilibrium measure and f is a smooth function], X(n)fnXfn(n) behaves like the eigenvalues counting function of the corresponding Wigner matrix and sample covariance matrix, respectively. MOE (Min. of Education, S’pore) Published version 2017-12-20T08:26:44Z 2019-12-06T16:27:42Z 2017-12-20T08:26:44Z 2019-12-06T16:27:42Z 2017 Journal Article Pan, G., Wang, S., & Zhou, W. (2017). Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory. Journal of Mathematical Physics, 58(10), 103301-. 0022-2488 https://hdl.handle.net/10356/86704 http://hdl.handle.net/10220/44176 10.1063/1.5006507 en Journal of Mathematical Physics © 2017 AIP Publishing. This paper was published in Journal of Mathematical Physics and is made available as an electronic reprint (preprint) with permission of AIP Publishing. The published version is available at: [https://doi.org/10.1063/1.5006507]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law. 15 p. application/pdf |
| spellingShingle | Probability Theory Correlation Pan, Guangming Wang, Shaochen Zhou, Wang Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory |
| title | Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory |
| title_full | Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory |
| title_fullStr | Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory |
| title_full_unstemmed | Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory |
| title_short | Limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory |
| title_sort | limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory |
| topic | Probability Theory Correlation |
| url | https://hdl.handle.net/10356/86704 http://hdl.handle.net/10220/44176 |
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