Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Comparison between two types of large sample covariance matrices

Guangming Pan

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Abstract

Let $\{X_{ij}\}$, $i,j=\cdots$, 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 $E|X_{11}|^{4}<\infty$. 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.

Résumé

Soit $\{X_{ij}\}$, $i,j=1,2,\ldots$, un tableau à double entrées, les $X_{ij}$ étant des variables aléatoires réelles indépendantes et identiquement distribuées (i.i.d.) et où $\mathbf{E} X_{11}=\mu$, $\mathbf{E} \vert X_{11}-\mu\vert ^{2}=1$ et $\mathbf{E} |X_{11}|^{4}<\infty$. Considérons les matrices de covariances empiriques suivantes (avec/sans centrage empirique): $\mathcal{S}=\frac{1}{n}\sum^{n}_{j=1}(\mathbf{s} _{j}-\bar{ \mathbf {s}})(\mathbf{s} _{j}-\bar{ \mathbf {s}})^{T}$ et $\mathbf{S}=\frac{1}{n}\sum^{n}_{j=1}\mathbf{s} _{j}\mathbf{s} _{j}^{T}$, avec $\bar{ \mathbf {s}}=\frac{1}{n}\sum^{n}_{j=1}\mathbf{s} _{j}$ et $\mathbf{s} _{j}=\mathbf{T}^{1/2}_{n}(X_{1j},\ldots,X_{pj})^{T}$, où $(\mathbf{T}^{1/2}_{n})^{2}=\mathbf{T}_{n}$ est une matrice déterministe définie positive. Nous démontrons que, sous le régime asymptotique $n\rightarrow\infty$ et $p/n$ converge vers une constante positive, le théorème central limite pour la statistique $\mathcal{S}$ est différent de celui concernant la statistique $\mathbf{S}$. En outre, nous montrons que cette différence de comportement n’est pas observée pour le comportement moyen des vecteurs propres.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 655-677.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1395856144

Digital Object Identifier
doi:10.1214/12-AIHP506

Mathematical Reviews number (MathSciNet)
MR3189088

Zentralblatt MATH identifier
1295.15023

Subjects
Primary: 15A52 60F15: Strong theorems 62E20: Asymptotic distribution theory 60F17: Functional limit theorems; invariance principles

Keywords
Central limit theorems Eigenvectors and eigenvalues Sample covariance matrix Stieltjes transform Strong convergence

Citation

Pan, Guangming. Comparison between two types of large sample covariance matrices. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 655--677. doi:10.1214/12-AIHP506. https://projecteuclid.org/euclid.aihp/1395856144


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