CLT For Linear Spectral Statistics Research Articles

CLT for linear spectral statistics of high-dimensional sample covariance matrices in elliptical distributions

In this paper, we establish a new central limit theorem for the linear spectral statistics of high-dimensional sample covariance matrices. The underlying population belongs to the family of elliptical distributions, and the dimension of the population is allowed to grow to infinity, in proportion to the sample size. As an application, we construct confidence intervals for the model parameters of a Gaussian scale mixture.

CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data

This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form \({\mathbf {B}}_n=n^{-1}\sum _{j=1}^{n}{\mathbf {Q}}{\mathbf {x}}_j{\mathbf {x}}_j^{*}{\mathbf {Q}}^{*}\) under the assumption that \(p/n\rightarrow y>0\), where \({\mathbf {Q}}\) is a \(p\times k\) nonrandom matrix and \(\{{\mathbf {x}}_j\}_{j=1}^n\) is a sequence of independent k-dimensional random vector with independent entries. A key novelty here is that the dimension \(k\ge p\) can be arbitrary, possibly infinity. This new model of sample covariance matrix \({\mathbf {B}}_n\) covers most of the known models as its special cases. For example, standard sample covariance matrices are obtained with \(k=p\) and \({\mathbf {Q}}={\mathbf {T}}_n^{1/2}\) for some positive definite Hermitian matrix \({\mathbf {T}}_n\). Also with \(k=\infty \) our model covers the case of repeated linear processes considered in recent high-dimensional time series literature. The CLT found in this paper substantially generalizes the seminal CLT in Bai and Silverstein (Ann Probab 32(1):553–605, 2004). Applications of this new CLT are proposed for testing the AR(1) or AR(2) structure for a causal process. Our proposed tests are then used to analyze a large fMRI data set.

A CLT for linear spectral statistics of large random information-plus-noise matrices

Consider a matrix Yn=σnXn+An, where σ>0 and Xn=(xijn) is a N×n random matrix with i.i.d. real or complex standardized entries and An is a N×n deterministic matrix with bounded spectral norm. The fluctuations of the linear spectral statistics of the eigenvalues: Tracef(YnYn∗)=∑i=1Nf(λi),(λi)eigenvalues ofYnYn∗,are shown to be Gaussian, in the case where f is a smooth function of class C3 with bounded support, and in the regime where both dimensions of matrix Yn go to infinity at the same pace. The CLT is expressed in terms of vanishing Lévy–Prokhorov distance between the linear statistics’ distribution and a centered Gaussian probability distribution, the variance of which depends upon N and n and may not converge. The proof combines ideas from [2,18] and [32].

Sharp detection in PCA under correlations: All eigenvalues matter

Principal component analysis (PCA) is a widely used method for dimension reduction. In high dimensional data, the "signal" eigenvalues corresponding to weak principal components (PCs) do not necessarily separate from the bulk of the "noise" eigenvalues. Therefore, popular tests based on the largest eigenvalue have little power to detect weak PCs. In the special case of the spiked model, certain tests asymptotically equivalent to linear spectral statistics (LSS)---averaging effects over all eigenvalues---were recently shown to achieve some power. We consider a nonparametric, non-Gaussian generalization of the spiked model to the setting of Marchenko and Pastur (1967). This allows a general bulk of the noise eigenvalues, accomodating correlated variables even under the null hypothesis of no significant PCs. We develop new tests based on LSS to detect weak PCs in this model. We show using the CLT for LSS that the optimal LSS satisfy a Fredholm integral equation of the first kind. We develop algorithms to solve it, building on our recent method for computing the limit empirical spectrum. In contrast to the standard spiked model, we find that under "widely spread" null eigenvalue distributions, the new tests have a lot of power.

Tests for large-dimensional covariance structure based on Rao’s score test

This paper proposes a new test for covariance matrices based on the correction to Rao’s score test in a large-dimension framework. By generalizing the corresponding CLT for linear spectral statistics, the test can be made applicable for large-dimension non-Gaussian variables in a wider range without the 4th-moment restriction. Moreover, the proposed corrected Rao’s score test (CRST) remains powerful even when p ≫ n , which breaks the inherent idea that the corrected tests by RMT can only be used when p < n . Finally, we compare the proposed test with other high-dimension covariance structure tests to evaluate their performances through a simulation study.

CLT for Linear Spectral Statistics of Hermitian Wigner Matrices with General Moment Conditions

In this paper, we study the Hermitian Wigner matrix $W_n=(x_{ij})_{1\le i,j\le n}$ with independent (up to symmetry) mean zero variance one entries. Under some Lindeberg type condition on the fourth moments of the entries, we establish a central limit theorem for the linear eigenvalue statistics of $W_n$. Our result extends the previous results on this topic to a more general case without the assumption ${\bf E}\,x_{ij}^2=0$ for $1\le i<j\le n$. Instead, we only assume that the real part and imaginary part of the upper-diagonal entry are uncorrelated. More precisely, we require ${\bf E}\,x_{ij}^2$ to be real and homogeneous for all $1\le i<j\le n$. The limiting normal distribution of the central limit theorem is shown to depend on the parameter ${\bf E}\,x_{ij}^2\in[-1,1]$.

CLT for linear spectral statistics of Hermitian Wigner matrices with general moment conditions

Изучается эрмитова матрица Вигнера $W_n=(x_{ij})_{1\leq i,j\leq n}$ с независимыми (с точностью до симметрии) внедиагональными элементами, имеющими нулевое среднее и единичную дисперсию. При некотором условии типа Линдеберга на четвертые моменты элементов матрицы доказана центральная предельная теорема для линейных статистик собственных значений матрицы $W_n$. Наш результат распространяет предыдущие результаты подобного рода на более общий случай, когда отсутствует ограничение $\bf E x^2_{ij}=0$ для $1\leq i &lt; j \leq n$. Вместо этого мы предполагаем только, что вещественная и мнимая части наддиагональных элементов некоррелированы. Более точно, мы требуем, чтобы математические ожидания $\bf E x^2_{ij}$ были вещественными и однородными для всех $1\leq i &lt; j \leq n$. Показано, что предельное нормальное распределение в центральной предельной теореме зависит от параметра $\bf E x^2_{ij}\in [-1,1]$.

On the sphericity test with large-dimensional observations

In this paper, we propose corrections to the likelihood ratio test and John’s test for sphericity in large-dimensions. New formulas for the limiting parameters in the CLT for linear spectral statistics of sample covariance matrices with general fourth moments are first established. Using these formulas, we derive the asymptotic distribution of the two proposed test statistics under the null. These asymptotics are valid for general population, i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive Monte-Carlo experiments are conducted to assess the quality of these tests with a comparison to several existing methods from the literature. Moreover, we also obtain their asymptotic power functions under the alternative of a spiked population model as a specific alternative.

CLT for Linear Spectral Statistics of Wigner matrices

In this paper, we prove that the spectral empirical process of Wigner matrices under sixth-moment conditions, which is indexed by a set of functions with continuous fourth-order derivatives on an open interval including the support of the semicircle law, converges weakly in finite dimensions to a Gaussian process.

CLT for linear spectral statistics of large-dimensional sample covariance matrices

Let $B_n=(1/N)T_n^{1/2}X_nX_n^*T_n^{1/2}$ where $X_n=(X_{ij})$ is $n\times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. The limiting behavior, as $n\to\infty$ with $n/N$ approaching a positive constant, of functionals of the eigenvalues of $B_n$, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of $B_n$, it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be $1/n$ by proving, after proper scaling, that they form a tight sequence. Moreover, if $\expp X^2_{11}=0$ and $\expp|X_{11}|^4=2$, or if $X_{11}$ and $T_n$ are real and $\expp X_{11}^4=3$, they are shown to have Gaussian limits.