Limiting spectral distribution of large-dimensional sample covariance matrices generated by VARMA

The existence of a limiting spectral distribution (LSD) for a large-dimensional sample covariance matrix generated by the vector autoregressive moving average (VARMA) model is established. In particular, we obtain explicit forms of the LSDs for random matrices generated by a first-order vector autoregressive (VAR(1)) model and a first-order vector moving average (VMA(1)) model, as well as random coefficients for VAR(1) and VMA(1). The parameters for these explicit forms are also estimated. Finally, simulations demonstrate that the results are effective.

[1]  Noureddine El Karoui,et al.  Operator norm consistent estimation of large-dimensional sparse covariance matrices , 2008, 0901.3220.

[2]  J. W. Silverstein.  Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices , 1995 .

[3]  P. Bickel,et al.  Covariance regularization by thresholding , 2009, 0901.3079.

[4]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of a class of large dimensional random matrices , 1995 .

[5]  E. Wigner.  Characteristic Vectors of Bordered Matrices with Infinite Dimensions I , 1955 .

[6]  P. Bickel,et al.  Regularized estimation of large covariance matrices , 2008, 0803.1909.

[7]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[8]  Z. Bai.  METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES , A REVIEW , 1999 .

[9]  P. Krishnaiah,et al.  A Limit Theorem for the Eigenvalues of Product of Two Random Matrices , 2003 .

[10]  J. W. Silverstein.  The Limiting Eigenvalue Distribution of a Multivariate F Matrix , 1985 .

[11]  E. Wigner.  On the Distribution of the Roots of Certain Symmetric Matrices , 1958 .

[12]  Y. Yin.  Limiting spectral distribution for a class of random matrices , 1986 .

[13]  Zhidong Bai,et al.  LARGE SAMPLE COVARIANCE MATRICES WITHOUT INDEPENDENCE STRUCTURES IN COLUMNS , 2008 .

[14]  Z. Bai,et al.  METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES, A REVIEW , 2008 .

[15]  Zhidong Bai,et al.  On limit theorem for the eigenvalues of product of two random matrices , 2007 .

[16]  J. W. Silverstein,et al.  Spectral Analysis of Large Dimensional Random Matrices , 2009 .

[17]  Z. D. Bai,et al.  On the Limiting Empirical Distribution Function of the Eigenvalues of a Multivariate F Matrix , 1988 .

[18]  Alexandre d'Aspremont,et al.  First-Order Methods for Sparse Covariance Selection , 2006, SIAM J. Matrix Anal. Appl..