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

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Abstract

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 $F$.

Résumé

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

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

Dates
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
https://projecteuclid.org/euclid.aihp/1479373243

Digital Object Identifier
doi:10.1214/15-AIHP693

Mathematical Reviews number (MathSciNet)
MR3573290

Zentralblatt MATH identifier
1359.15034

Subjects
Primary: 15A52 60F15: Strong theorems

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

Citation

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. https://projecteuclid.org/euclid.aihp/1479373243


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