Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Moment approach for singular values distribution of a large auto-covariance matrix

Qinwen Wang and Jianfeng Yao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $(\varepsilon_{t})_{t>0}$ be a sequence of independent real random vectors of $p$-dimension and let $X_{T}=\sum_{t=s+1}^{s+T}\varepsilon_{t}\varepsilon^{*}_{t-s}/T$ be the lag-$s$ ($s$ is a fixed positive integer) auto-covariance matrix of $\varepsilon_{t}$. Since $X_{T}$ is not symmetric, we consider its singular values, which are the square roots of the eigenvalues of $X_{T}X^{*}_{T}$. Using the method of moments, we are able to investigate the limiting behaviors of the eigenvalues of $X_{T}X^{*}_{T}$ in two aspects. First, we show that the empirical spectral distribution of its eigenvalues converges to a nonrandom limit $F$, which is a result previously developed in (J. Multivariate Anal. 137 (2015) 119–140) using the Stieltjes transform method. Second, we establish the convergence of its largest eigenvalue to the right edge of $F$.


Soit $(\varepsilon_{t})_{t>0}$ une suite de vecteurs aléatoires indépendants de $\mathbb{R}^{p}$ et $X_{T}=\sum_{t=s+1}^{s+t}\varepsilon_{t}\varepsilon_{t-s}^{*}/T$ la matrice d’autocovariance empirique d’ordre $s$ de la suite ($s$ est un ordre fixé). Comme $X_{T}$ n’est pas symétrique, nous considérons ses valeurs singulières, c’est-à-dire les racines carrées des valeurs propres de la matrice aléatoire $X_{T}X_{T}^{*}$. En utilisant la méthode des moments, nous établissons les propriétés limites de ces valeurs singulières dans deux directions. D’abord, nous démontrons que leur distribution empirique converge vers une limite déterministe $F$, retrouvant ainsi un résultat établi dans (J. Multivariate Anal. 137 (2015) 119–140) par la méthode de la transformée de Stieltjes. Ensuite, nous montrons que la plus grande de ces valeurs singulières converge vers le point extrémal du support de $F$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1641-1666.

Received: 28 March 2015
Revised: 3 June 2015
Accepted: 15 June 2015
First available in Project Euclid: 17 November 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52 60F15: Strong theorems

Auto-covariance matrix Singular values Limiting spectral distribution Stieltjes transform Largest eigenvalue Moment method


Wang, Qinwen; Yao, Jianfeng. Moment approach for singular values distribution of a large auto-covariance matrix. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1641--1666. doi:10.1214/15-AIHP693.

Export citation


  • [1] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices, 2nd edition. Springer, New York, 2010.
  • [2] Z. D. Bai and Y. Q. Yin. Necessary and sufficient conditions for the almost sure convergence of the largest eigenvalue of Wigner matrices. Ann. Probab. 16 (4) (1988) 1729–1741.
  • [3] S. Geman. A limit theorem for the norm of random matrices. Ann. Probab. 8 (1980) 252–261.
  • [4] V. L. Girko. Circle law. Theory Probab. Appl. 4 (1984) 694–706.
  • [5] P. Hilton and J. Pedersen. Catalan numbers, their generalization, and their uses. Math. Intelligencer 13 (1991) 64–75.
  • [6] B. S. Jin, C. Wang, Z. D. Bai, K. K. Nair and M. C. Harding. Limiting spectral distribution of a symmetrized auto-cross covariance matrix. Ann. Appl. Probab. 24 (2014) 1199–1225.
  • [7] C. Lam and Q. W. Yao. Factor modeling for high-dimensional time series: Inference for the number of factors. Ann. Statist. 40 (2012) 694–726.
  • [8] Z. Li, G. M. Pan and J. F. Yao. On singular value distribution of large-dimensional autocovariance matrices. J. Multivariate Anal. 137 (2015) 119–140.
  • [9] H. Y. Liu, A. Aue and D. Paul. On the Marčenko-Pastur law for linear time series. Ann. Statist. 43 (2015) 675–712.
  • [10] V. A. Marčenko and L. A. Pastur. Distribution of eigenvalues for some sets of random matrices. Mathematics of the USSR. Sbornik 1 (1967) 457–483.
  • [11] S. Péché and A. Soshnikov. On the lower bound of the spectral norm of symmetric random matrices with independent entries. Electron. Commun. Probab. 13 (2008) 280–290.
  • [12] G. Pólya and G. Szegö. Problems and Theorems in Analysis (Volume I). Springer, New York, 1976.
  • [13] T. Tao. Topics in Random Matrix Theory. Amer. Math. Soc., Providence, RI, 2012.
  • [14] R. Vershynin. Spectral norm of products of random and deterministic matrices. Probab. Theory Related Fields 150 (2015) 471–509.
  • [15] V. Vu. Spectral norm of random matrices. Combinatorica 27 (6) (2007) 721–736.
  • [16] C. Wang, B. S. Jin, Z. D. Bai, K. K. Nair and M. C. Harding. Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix. Preprint, 2013. Available at arXiv:1312.2277.
  • [17] Q. W. Wang and J. F. Yao. On singular values distribution of a large auto-covariance matrix in the ultra-dimensional regime. Preprint, 2015. Available at arXiv:1501.06641.
  • [18] E. P. Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 (1958) 325–327.
  • [19] Y. Q. Yin, Z. D. Bai and P. R. Krishnaiah. On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Related Fields 78 (1988) 509–521.