THE EMPIRICAL EIGENVALUE DISTRIBUTION OF A GRAM MATRIX: FROM INDEPENDENCE TO STATIONARITY

By Walid Hachem, Philippe Loubaton, Jamal Najim. 2005

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Consider a N × n matrix Zn = (Zn j1j2 ) where the individual entries are a realization of a properly rescaled stationary gaussian random field: Z j1j2 = 1 √ n ∑ (k1,k2)∈Z2 h(k1, k2)U(j1 − k1, j2 − k2), where h ∈ `1(Z2) is a deterministic complex summable sequence and (U(j1, j2); (j1, j2) ∈ Z2) is a sequence of independent complex gaussian random variables with mean zero and unit variance. The purpose of this article is to study the limiting empirical distribution of the eigenvalues of Gram random matrices such as ZnZ∗ n and (Zn + An)(Zn + An) ∗ where An is a deterministic matrix with appropriate assumptions in the case where n → ∞ and N n → c ∈ (0,∞). The proof relies on related results for matrices with independent but not identically distributed entries and substantially differs from related works in the literature (Boutet de Monvel et al. [3], Girko [7], etc.).

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/818df8e292099182e52e34664c91ae013e426843

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Citations in the corpus

On the Marčenko-Pastur law for linear time series (2013, background): “[16], but additional technical arguments are needed to go beyond Gaussanity.”
Exponent of Cross-Sectional Dependence: Estimation and Inference (2016, unknown): “Accordingly, Chudik, Pesaran, and Tosetti (2011), Kapetanios and Marcellino (2010) and Onatski (2011) have considered an extension of the above factor model which allows for a wider spectrum of cross-sectional dependence behaviours by specifying that the factor loadings βi may be functions of N and ”
Random matrix theory in statistics: A review (2014, methodology): “Similar ideas have been used by Hachem et al. (2005) to derive the limiting ESD of n 1YnYnn when the entries of Yn have the moving average representation Yjk ¼∑ðl;l′ÞAZ2hl;l′Zl j;l′ k, where Zl;l′ are i.i.d. complex Gaussian with zero mean and unit variance, and the deterministic complex sequence…”
A Testing Procedure for Determining the Number of Factors in Approximate Factor Models With Large Datasets (2010, background): “Recently, work by Hachem, Loubaton, and Najim (2005a, 2005b) derived the limit of the ESD of the sample covariance matrix of temporally independent, but heterogeneously distributed data and temporally dependent data with absolutely summable autocovariances.”
A CLT FOR INFORMATION-THEORETIC STATISTICS OF GRAM RANDOM MATRICES WITH A GIVEN VARIANCE PROFILE (2007, unknown): “ study of the convergence and the ﬂuctuations of the mutual information 1 N logdet( Yn ∗ n +ρIN) for various models of matrices Yn. General convergence results have been established by the authors in [17, 15, 16] while ﬂuctuation results based on Bai and Silverstein [3] have been developed in Debbah”
The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile (2004, background): “Links between Gram matrices with a variance profile and Gram matrices based on Gaussian fields are studied in [11].”
Determining the Number of Factors from Empirical Distribution of Eigenvalues (2010, unknown): “Therefore, (21) holds with ζ = Op 3 n−β1+ 1 3 (1−α) ́ .”
A law of large numbers for finite‐range dependent random matrices (2006, methodology): “They also apply to the equations for the limit of the Stieltjes tranform in [ HLN05 ] and the systems of equations in [Gi90, Corollaries 9.3.2 and 10.1.2], under appropriate hypotheses of “finiteness of filter”, as well as many of the pre-limit “Kx cannonical equations” in [Gi01], x2 {1,...,30}, and”