Comparison between two types of large sample covariance matrices
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 655-677.

Let $\left\{{X}_{ij}\right\}$, $i,j=\cdots$, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu$, $E|{X}_{11}{-\mu |}^{2}=1$ and $E|{X}_{11}{|}^{4}<\infty$. Consider sample covariance matrices (with/without empirical centering) $𝒮=\frac{1}{n}{\sum }_{j=1}^{n}\left({𝐬}_{j}-\overline{𝐬}\right){\left({𝐬}_{j}-\overline{𝐬}\right)}^{T}$ and $𝐒=\frac{1}{n}{\sum }_{j=1}^{n}{𝐬}_{j}{𝐬}_{j}^{T}$, where $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^{n}{𝐬}_{j}$ and ${𝐬}_{j}={𝐓}_{n}^{1/2}{\left({X}_{1j},...,{X}_{pj}\right)}^{T}$ with ${\left({𝐓}_{n}^{1/2}\right)}^{2}={𝐓}_{n}$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $𝒮$ and $𝐒$ are different as $n\to \infty$ 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.

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

DOI: 10.1214/12-AIHP506
Classification: 15A52,  60F15,  62E20,  60F17
Keywords: central limit theorems, eigenvectors and eigenvalues, sample covariance matrix, Stieltjes transform, strong convergence
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title = {Comparison between two types of large sample covariance matrices},
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Pan, Guangming. Comparison between two types of large sample covariance matrices. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 655-677. doi : 10.1214/12-AIHP506. http://www.numdam.org/articles/10.1214/12-AIHP506/

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