CLT for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size

Let $\mathbf{A}=\frac{1}{\sqrt{np}}(\mathbf{X}^T\mathbf{X}-p\mathbf {I}_n)$ where $\mathbf{X}$ is a $p\times n$ matrix, consisting of independent and identically distributed (i.i.d.) real random variables $X_{ij}$ with mean zero and variance one. When $p/n\to\infty$, under fourth moment conditions a central limit theorem (CLT) for linear spectral statistics (LSS) of $\mathbf{A}$ defined by the eigenvalues is established. We also explore its applications in testing whether a population covariance matrix is an identity matrix.Comment: Published at http://dx.doi.org/10.3150/14-BEJ599 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Independence test for high dimensional data based on regularized canonical correlation coefficients

This paper proposes a new statistic to test independence between two high dimensional random vectors ${\mathbf{X}}:p_1\times1$ and ${\mathbf{Y}}:p_2\times1$. The proposed statistic is based on the sum of regularized sample canonical correlation coefficients of ${\mathbf{X}}$ and ${\mathbf{Y}}$. The asymptotic distribution of the statistic under the null hypothesis is established as a corollary of general central limit theorems (CLT) for the linear statistics of classical and regularized sample canonical correlation coefficients when $p_1$ and $p_2$ are both comparable to the sample size $n$. As applications of the developed independence test, various types of dependent structures, such as factor models, ARCH models and a general uncorrelated but dependent case, etc., are investigated by simulations. As an empirical application, cross-sectional dependence of daily stock returns of companies between different sections in the New York Stock Exchange (NYSE) is detected by the proposed test.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1284 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Universality for the largest eigenvalue of sample covariance matrices with general population

This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form $\mathcal{W}_N=\Sigma^{1/2}XX^*\Sigma ^{1/2}$. Here, $X=(x_{ij})_{M,N}$ is an $M\times N$ random matrix with independent entries $x_{ij},1\leq i\leq M,1\leq j\leq N$ such that $\mathbb{E}x_{ij}=0$, $\mathbb{E}|x_{ij}|^2=1/N$. On dimensionality, we assume that $M=M(N)$ and $N/M\rightarrow d\in(0,\infty)$ as $N\rightarrow\infty$. For a class of general deterministic positive-definite $M\times M$ matrices $\Sigma$, under some additional assumptions on the distribution of $x_{ij}$'s, we show that the limiting behavior of the largest eigenvalue of $\mathcal{W}_N$ is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd\H{o}s, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case ($\Sigma=I$). Consequently, in the standard complex case ($\mathbb{E}x_{ij}^2=0$), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the largest eigenvalue of $\mathcal{W}_N$ converges weakly to the type 2 Tracy-Widom distribution $\mathrm{TW}_2$. Moreover, in the real case, we show that when $\Sigma$ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit $\mathrm{TW}_1$ holds for the normalized largest eigenvalue of $\mathcal {W}_N$, which extends a result of F\'{e}ral and P\'{e}ch\'{e} in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal $\Sigma$ and more generally distributed $X$.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1281 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Universality for a global property of the eigenvectors of Wigner matrices

Let $M_n$ be an $n\times n$ real (resp. complex) Wigner matrix and $U_n\Lambda_n U_n^*$ be its spectral decomposition. Set $(y_1,y_2...,y_n)^T=U_n^*x$, where $x=(x_1,x_2,...,$ $x_n)^T$ is a real (resp. complex) unit vector. Under the assumption that the elements of $M_n$ have 4 matching moments with those of GOE (resp. GUE), we show that the process $X_n(t)=\sqrt{\frac{\beta n}{2}}\sum_{i=1}^{\lfloor nt\rfloor}(|y_i|^2-\frac1n)$ converges weakly to the Brownian bridge for any $\mathbf{x}$ such that $||x||_\infty\rightarrow 0$ as $n\rightarrow \infty$, where $\beta=1$ for the real case and $\beta=2$ for the complex case. Such a result indicates that the othorgonal (resp. unitary) matrices with columns being the eigenvectors of Wigner matrices are asymptotically Haar distributed on the orthorgonal (resp. unitary) group from a certain perspective.Comment: typos correcte

On singular value distribution of large dimensional auto-covariance matrices

Let $(\varepsilon_j)_{j\geq 0}$ be a sequence of independent $p-$dimensional random vectors and $\tau\geq1$ a given integer. From a sample $\varepsilon_1,\cdots,\varepsilon_{T+\tau-1},\varepsilon_{T+\tau}$ of the sequence, the so-called lag $-\tau$ auto-covariance matrix is $C_{\tau}=T^{-1}\sum_{j=1}^T\varepsilon_{\tau+j}\varepsilon_{j}^t$. When the dimension $p$ is large compared to the sample size $T$, this paper establishes the limit of the singular value distribution of $C_\tau$ assuming that $p$ and $T$ grow to infinity proportionally and the sequence satisfies a Lindeberg condition on fourth order moments. Compared to existing asymptotic results on sample covariance matrices developed in random matrix theory, the case of an auto-covariance matrix is much more involved due to the fact that the summands are dependent and the matrix $C_\tau$ is not symmetric. Several new techniques are introduced for the derivation of the main theorem

Tracy-Widom law for the extreme eigenvalues of sample correlation matrices

Let the sample correlation matrix be $W=YY^T$, where $Y=(y_{ij})_{p,n}$ with $y_{ij}=x_{ij}/\sqrt{\sum_{j=1}^nx_{ij}^2}$. We assume $\{x_{ij}: 1\leq i\leq p, 1\leq j\leq n\}$ to be a collection of independent symmetric distributed random variables with sub-exponential tails. Moreover, for any $i$, we assume $x_{ij}, 1\leq j\leq n$ to be identically distributed. We assume $0<p<n$ and $p/n\rightarrow y$ with some $y\in(0,1)$ as $p,n\rightarrow\infty$. In this paper, we provide the Tracy-Widom law ($TW_1$) for both the largest and smallest eigenvalues of $W$. If $x_{ij}$ are i.i.d. standard normal, we can derive the $TW_1$ for both the largest and smallest eigenvalues of the matrix $\mathcal{R}=RR^T$, where $R=(r_{ij})_{p,n}$ with $r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$.Comment: 35 pages, a major revisio

Comparison between two types of large sample covariance matrices

Let {Xij}, i, j = · · · , be a double array of independent and identically distributed (i.i.d.) real random variables with EX11= μ, E|X11 − μ|2 = 1 and E|X11|4 < ∞. Consider sample covariance matrices (with/without empirical centering) S = 1/n nΣj=1 (sj− s)(sj −¯s)T and S = 1/n nΣj=1 sjsTj, where ¯s =1/n nΣj=1 sj and sj = T1/2 n (X1j , · · · ,Xpj)T with (T1/2 n )2 = Tn, 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 → ∞ 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.Accepted versio