Comparison between two types of large sample covariance matrices

Pan, Guangming

Pan, Guangming

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 655-677 / Harvested from Numdam

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Résumé
Abstract

Soit ${X_{i j}}$ , $i, j = 1, 2, ...$ , un tableau à double entrées, les $X_{i j}$ étant des variables aléatoires réelles indépendantes et identiquement distribuées (i.i.d.) et où $𝐄 X_{11} = μ$ , $𝐄 | X_{11} {- μ |}^{2} = 1$ et $𝐄 | X_{11} |^{4} < \infty$ . Considérons les matrices de covariances empiriques suivantes (avec/sans centrage empirique): $𝒮 = \frac{1}{n} \sum_{j = 1}^{n} (𝐬_{j} - \bar{𝐬}) {(𝐬_{j} - \bar{𝐬})}^{T}$ et $𝐒 = \frac{1}{n} \sum_{j = 1}^{n} 𝐬_{j} 𝐬_{j}^{T}$ , avec $\bar{𝐬} = \frac{1}{n} \sum_{j = 1}^{n} 𝐬_{j}$ et $𝐬_{j} = 𝐓_{n}^{1 / 2} {(X_{1 j}, ..., X_{p j})}^{T}$ , où ${(𝐓_{n}^{1 / 2})}^{2} = 𝐓_{n}$ 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.

Let ${X_{i j}}$ , $i, j = \dots$ , 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 $E | X_{11} |^{4} < \infty$ . Consider sample covariance matrices (with/without empirical centering) $𝒮 = \frac{1}{n} \sum_{j = 1}^{n} (𝐬_{j} - \bar{𝐬}) {(𝐬_{j} - \bar{𝐬})}^{T}$ and $𝐒 = \frac{1}{n} \sum_{j = 1}^{n} 𝐬_{j} 𝐬_{j}^{T}$ , where $\bar{𝐬} = \frac{1}{n} \sum_{j = 1}^{n} 𝐬_{j}$ 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 $n \to \infty$ 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.

Publié le : 2014-01-01

MR 3189088 | Zbl 1295.15023

DOI : https://doi.org/10.1214/12-AIHP506
Classification: 15A52, 60F15, 62E20, 60F17

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     author = {Pan, Guangming},
     title = {Comparison between two types of large sample covariance matrices},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {50},
     year = {2014},
     pages = {655-677},
     doi = {10.1214/12-AIHP506},
     mrnumber = {3189088},
     zbl = {1295.15023},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2014__50_2_655_0}
}

Pan, Guangming. Comparison between two types of large sample covariance matrices. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) pp. 655-677. doi : 10.1214/12-AIHP506. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_2_655_0/

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