Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
Let the sample correlation matrix be W = YYT, where Y = (yij)p,n with yij = xij /√∑xij2. We assume {xij : 1 ≤ i ≤ p, 1 ≤ j ≤ n} to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any i, we assume xij, 1 ≤ j ≤ n to be identically dis...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2013
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| Online Access: | https://hdl.handle.net/10356/96096 http://hdl.handle.net/10220/10085 |
| _version_ | 1811683673807781888 |
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| author | Bao, Zhigang Pan, Guangming Zhou, Wang |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Bao, Zhigang Pan, Guangming Zhou, Wang |
| author_sort | Bao, Zhigang |
| collection | NTU |
| description | Let the sample correlation matrix be W = YYT, where Y = (yij)p,n with yij = xij /√∑xij2. We assume {xij : 1 ≤ i ≤ p, 1 ≤ j ≤ n} to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any i, we assume xij, 1 ≤ j ≤ n to be identically distributed. We assume 0 < p < n and p/n→y with some y ∈ (0,1) as p,n → ∞. In this paper, we provide the Tracy-Widom law (TW1) for both the largest and smallest eigenvalues of W. If xij are i.i.d. standard normal, we can derive the TW1 for both the largest and smallest eigenvalues of the matrix R = RRT, where R = (rij)p,n with rij = (xij − xi )/√∑(xij −xi)2, xi = n−1∑xij. |
| first_indexed | 2024-10-01T04:16:29Z |
| format | Journal Article |
| id | ntu-10356/96096 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T04:16:29Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | ntu-10356/960962023-02-28T19:22:39Z Tracy-Widom law for the extreme eigenvalues of sample correlation matrices Bao, Zhigang Pan, Guangming Zhou, Wang School of Physical and Mathematical Sciences Let the sample correlation matrix be W = YYT, where Y = (yij)p,n with yij = xij /√∑xij2. We assume {xij : 1 ≤ i ≤ p, 1 ≤ j ≤ n} to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any i, we assume xij, 1 ≤ j ≤ n to be identically distributed. We assume 0 < p < n and p/n→y with some y ∈ (0,1) as p,n → ∞. In this paper, we provide the Tracy-Widom law (TW1) for both the largest and smallest eigenvalues of W. If xij are i.i.d. standard normal, we can derive the TW1 for both the largest and smallest eigenvalues of the matrix R = RRT, where R = (rij)p,n with rij = (xij − xi )/√∑(xij −xi)2, xi = n−1∑xij. Published version 2013-06-10T02:27:08Z 2019-12-06T19:25:36Z 2013-06-10T02:27:08Z 2019-12-06T19:25:36Z 2012 2012 Journal Article Bao, Z., Pan, G., & Zhou, W. (2012). Tracy-Widom law for the extreme eigenvalues of sample correlation matrices. Electronic Journal of Probability, 17(88), 1-32. 1083-6489 https://hdl.handle.net/10356/96096 http://hdl.handle.net/10220/10085 10.1214/EJP.v17-1962 en Electronic journal of probability © 2012 The Authors. This paper was published in Electronic Journal of Probability and is made available as an electronic reprint (preprint) with permission of The Authors. The paper can be found at the following official DOI: [http://dx.doi.org/10.1214/EJP.v17-1962]. 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. application/pdf |
| spellingShingle | Bao, Zhigang Pan, Guangming Zhou, Wang Tracy-Widom law for the extreme eigenvalues of sample correlation matrices |
| title | Tracy-Widom law for the extreme eigenvalues of sample correlation matrices |
| title_full | Tracy-Widom law for the extreme eigenvalues of sample correlation matrices |
| title_fullStr | Tracy-Widom law for the extreme eigenvalues of sample correlation matrices |
| title_full_unstemmed | Tracy-Widom law for the extreme eigenvalues of sample correlation matrices |
| title_short | Tracy-Widom law for the extreme eigenvalues of sample correlation matrices |
| title_sort | tracy widom law for the extreme eigenvalues of sample correlation matrices |
| url | https://hdl.handle.net/10356/96096 http://hdl.handle.net/10220/10085 |
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