Comparison between two types of large sample covariance matrices

Guangming Pan

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 2, page 655-677
  • ISSN: 0246-0203

Abstract

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Let { X i j } , i , j = ⋯ , be a double array of independent and identically distributed (i.i.d.) real random variables with E X 11 = μ , E | X 11 - μ | 2 = 1 and E | X 11 | 4 l t ; ∞ . Consider sample covariance matrices (with/without empirical centering) 𝒮 = 1 n ∑ j = 1 n ( 𝐬 j - 𝐬 ¯ ) ( 𝐬 j - 𝐬 ¯ ) T and 𝐒 = 1 n ∑ j = 1 n 𝐬 j 𝐬 j T , where 𝐬 ¯ = 1 n ∑ j = 1 n 𝐬 j and 𝐬 j = 𝐓 n 1 / 2 ( X 1 j , ... , X p j ) T with ( 𝐓 n 1 / 2 ) 2 = 𝐓 n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of 𝒮 and 𝐒 are different as n → ∞ with p / n approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior of eigenvectors.

How to cite

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Pan, Guangming. "Comparison between two types of large sample covariance matrices." Annales de l'I.H.P. Probabilités et statistiques 50.2 (2014): 655-677. <http://eudml.org/doc/271990>.

@article{Pan2014,
abstract = {Let $\lbrace X_\{ij\}\rbrace $, $i,j=\cdots $, be a double array of independent and identically distributed (i.i.d.) real random variables with $EX_\{11\}=\mu $, $E|X_\{11\}-\mu |^\{2\}=1$ and $E|X_\{11\}|^\{4\}&lt;\infty $. Consider sample covariance matrices (with/without empirical centering) $\mathcal \{S\}=\frac\{1\}\{n\}\sum _\{j=1\}^\{n\}(\mathbf \{s\}_\{j\}-\bar\{\mathbf \{s\}\})(\mathbf \{s\}_\{j\}-\bar\{\mathbf \{s\}\})^\{T\}$ and $\mathbf \{S\} =\frac\{1\}\{n\}\sum _\{j=1\}^\{n\}\mathbf \{s\}_\{j\}\mathbf \{s\}_\{j\}^\{T\}$, where $\bar\{\mathbf \{s\}\}=\frac\{1\}\{n\}\sum _\{j=1\}^\{n\}\mathbf \{s\}_\{j\}$ and $\mathbf \{s\}_\{j\}=\mathbf \{T\} _\{n\}^\{1/2\}(X_\{1j\},\ldots ,X_\{pj\})^\{T\}$ with $(\mathbf \{T\} _\{n\}^\{1/2\})^\{2\}=\mathbf \{T\} _\{n\}$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $\mathcal \{S\}$ and $\mathbf \{S\} $ are different as $n\rightarrow \infty $ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior of eigenvectors.},
author = {Pan, Guangming},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {central limit theorems; eigenvectors and eigenvalues; sample covariance matrix; Stieltjes transform; strong convergence; eigenvectors; eigenvalues; eigenvalue statistics},
language = {eng},
number = {2},
pages = {655-677},
publisher = {Gauthier-Villars},
title = {Comparison between two types of large sample covariance matrices},
url = {http://eudml.org/doc/271990},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Pan, Guangming
TI - Comparison between two types of large sample covariance matrices
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 2
SP - 655
EP - 677
AB - Let $\lbrace X_{ij}\rbrace $, $i,j=\cdots $, be a double array of independent and identically distributed (i.i.d.) real random variables with $EX_{11}=\mu $, $E|X_{11}-\mu |^{2}=1$ and $E|X_{11}|^{4}&lt;\infty $. Consider sample covariance matrices (with/without empirical centering) $\mathcal {S}=\frac{1}{n}\sum _{j=1}^{n}(\mathbf {s}_{j}-\bar{\mathbf {s}})(\mathbf {s}_{j}-\bar{\mathbf {s}})^{T}$ and $\mathbf {S} =\frac{1}{n}\sum _{j=1}^{n}\mathbf {s}_{j}\mathbf {s}_{j}^{T}$, where $\bar{\mathbf {s}}=\frac{1}{n}\sum _{j=1}^{n}\mathbf {s}_{j}$ and $\mathbf {s}_{j}=\mathbf {T} _{n}^{1/2}(X_{1j},\ldots ,X_{pj})^{T}$ with $(\mathbf {T} _{n}^{1/2})^{2}=\mathbf {T} _{n}$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $\mathcal {S}$ and $\mathbf {S} $ are different as $n\rightarrow \infty $ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior of eigenvectors.
LA - eng
KW - central limit theorems; eigenvectors and eigenvalues; sample covariance matrix; Stieltjes transform; strong convergence; eigenvectors; eigenvalues; eigenvalue statistics
UR - http://eudml.org/doc/271990
ER -

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