Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero
Let Xp = (s1, . . . , sn) = (Xij )p×n where Xij ’s are independent and identically distributed (i.i.d.) random variables with EX11 = 0, EX2 11 = 1 and EX4 11 <1. It is showed that the largest eigen- value of the random matrix Ap = 1 2√np (XpX′p −nIp) tends to 1 almost surely as p→∞,n→∞ with p/n→0...
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sg-ntu-dr.10356-988642023-02-28T19:22:56Z Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero Chen, B. B. Pan, G. M. School of Physical and Mathematical Sciences Let Xp = (s1, . . . , sn) = (Xij )p×n where Xij ’s are independent and identically distributed (i.i.d.) random variables with EX11 = 0, EX2 11 = 1 and EX4 11 <1. It is showed that the largest eigen- value of the random matrix Ap = 1 2√np (XpX′p −nIp) tends to 1 almost surely as p→∞,n→∞ with p/n→0. Published Version 2013-08-01T01:06:17Z 2019-12-06T20:00:36Z 2013-08-01T01:06:17Z 2019-12-06T20:00:36Z 2012 2012 Journal Article Chen, B. B., & Pan, G. M. (2012). Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero. Bernoulli, 18(4), 1405-1420. 1350-7265 https://hdl.handle.net/10356/98864 http://hdl.handle.net/10220/12678 10.3150/11-BEJ381 en Bernoulli © 2012 ISI/BS. This paper was published in Bernoulli and is made available as an electronic reprint (preprint) with permission of Bernoulli Society for Mathematical Statistics and Probability. The paper can be found at the following official DOI: [http://dx.doi.org/10.3150/11-BEJ381]. 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 |
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Let Xp = (s1, . . . , sn) = (Xij )p×n where Xij ’s are independent and identically distributed (i.i.d.) random variables with EX11 = 0, EX2 11 = 1 and EX4 11 <1. It is showed that the largest eigen- value of the random matrix Ap = 1 2√np (XpX′p −nIp) tends to 1 almost surely as p→∞,n→∞ with p/n→0. |
| author2 |
School of Physical and Mathematical Sciences |
| author_facet |
School of Physical and Mathematical Sciences Chen, B. B. Pan, G. M. |
| format |
Article |
| author |
Chen, B. B. Pan, G. M. |
| spellingShingle |
Chen, B. B. Pan, G. M. Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero |
| author_sort |
Chen, B. B. |
| title |
Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero |
| title_short |
Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero |
| title_full |
Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero |
| title_fullStr |
Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero |
| title_full_unstemmed |
Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero |
| title_sort |
convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero |
| publishDate |
2013 |
| url |
https://hdl.handle.net/10356/98864 http://hdl.handle.net/10220/12678 |
| _version_ |
1759856345471778816 |