A note on the convergence rate of the spectral distributions of large random matrices

Bai, Zhidong; Bai-qi, Miao; Tsay, Jhishen

doi:10.1016/s0167-7152(96)00170-8

Cited by 22 publications

(22 citation statements)

References 3 publications

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“…Applying this inequality, a convergence rate for the expected ESD of a large Wigner matrix was proved to be O(n −1/4 ) and that for the sample covariance matrix was shown to be O(n −1/4 ) if the ratio of the dimension to the degrees of freedom is far from 1 and O(n −5/48 ) if the ratio is close to 1. Some further developments can be found in Bai et al [23,24,25], Bai et al [26], Götze et al [132], and Götze and Tikhomirov [133,134].…”

Section: Convergence Rate Of the Esdmentioning

confidence: 92%

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Spectral Analysis of Large Dimensional Random Matrices

Bai

Silverstein

2010

Springer Series in Statistics

1,339

2,100

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Section: Convergence Rate Of the Esdmentioning

confidence: 92%

“…25. Suppose that {X jk } are iid complex random variables with EX 11 = 0, E|X 2 11 | = 1, and E|X 4 11 | < ∞.…”

Section: New Developmentsmentioning

confidence: 99%

Spectral Analysis of Large Dimensional Random Matrices

Bai

Silverstein

2010

Springer Series in Statistics

1,339

2,100

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“…Here, we present a new result of Bai, Miao and Tsay (1997) in which a convergence rate in probability is established. Readers interested in the details of the proof of Theorem 3.6 are referred to Bai (1993a).…”

Section: Wigner Matrixmentioning

confidence: 99%

Methodologies in Spectral Analysis of Large Dimensional Random Matrices, a Review

Bai

2008

Advances in Statistics

410

600

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In this paper, we give a brief review of the theory of spectral analysis of large dimensional random matrices. Most of the existing work in the literature has been stated for real matrices but the corresponding results for the complex case are also of interest, especially for researchers in Electrical and Electronic Engineering. Thus, we convert almost all results to the complex case, whenever possible. Only the latest results, including some new ones, are stated as theorems here. The main purpose of the paper is to show how important methodologies, or mathematical tools, have helped to develop the theory. Some unsolved problems are also stated.Key words and phrases: Circular law, complex random matrix, largest and smallest eigenvalues of a random matrix, noncentral Hermitian matrix, spectral analysis of large dimensional random matrices, spectral radius.

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“…Applying this inequality, a convergence rate for the expected ESD of a large Wigner matrix was proved to be Oðn À 1=4 Þ and that for the sample covariance matrix was shown to be Oðn À 1=4 Þ if the ratio of the dimension to the degrees of freedom is far from 1, and Oðn À 5=48 Þ if the ratio is close to 1. Some further developments can be found in Bai et al (1997. In Bai et al (2003), the rate of convergence for the ESD of a sample covariance matrix when γ ¼ 1 was improved to Oðn À 1=8 Þ.…”

Section: Convergence Rate Of the Esdmentioning

confidence: 99%

Random matrix theory in statistics: A review

Paul

Aue

2014

Journal of Statistical Planning and Inference

165

115

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b s t r a c tWe give an overview of random matrix theory (RMT) with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies. This paper focuses on a number of application areas especially within the field of high-dimensional statistics and describes how the development of the theory and practice in high-dimensional statistical inference has been influenced by the corresponding developments in the field of RMT.

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A note on the convergence rate of the spectral distributions of large random matrices

Cited by 22 publications

References 3 publications

Spectral Analysis of Large Dimensional Random Matrices

Spectral Analysis of Large Dimensional Random Matrices

Methodologies in Spectral Analysis of Large Dimensional Random Matrices, a Review

Random matrix theory in statistics: A review

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