Search

Consider a complex random $n\times n$ matrix ${\bf X}_n=(x_{ij})_{n\times n}$, whose entries $x_{ij}$ are independent random variables with zero means and unit variances. It is well-known that Tao and Vu (Ann Probab 38: 2023-2065, 2010) resolved the circular law conjecture, establishing that if the $x_{ij}$'s are independent and identically distributed random variables with zero mean and unit variance, the empirical spectral distribution of $\frac{1}{\sqrt{n}}{\bf X}_n$ converges almost surely to the uniform distribution over the unit disk in the complex plane as $n \to \infty$. This paper demonstrates that the circular law still holds under the more general Lindeberg's condition: $$ \frac1{n^2}\sum_{i,j=1}^n\mathbb{E}|x_{ij}^2|I(|x_{ij}|>\eta\sqrt{n})\to 0,\mbox{as $n \to \infty$}. $$ This paper is a revisit of the proof procedure of the circular law by Bai in (Ann Probab 25: 494-529, 1997). The key breakthroughs in the paper are establishing a general strong law of large numbers under Lindeberg's condition and the uniform upper bound for the integral with respect to the smallest eigenvalues of random matrices. These advancements significantly streamline and clarify the proof of the circular law, offering a more direct and simplified approach than other existing methodologies.

The random self-reinforcement mechanism, characterized by the principle of ``the rich get richer'', has demonstrated significant utility across various domains. One prominent model embodying this mechanism is the random reinforcement urn model. This paper investigates a multicolored, multiple-drawing variant of the random reinforced urn model. We establish the limiting behavior of the normalized urn composition and demonstrate strong convergence upon scaling the counts of each color. Additionally, we derive strong convergence estimators for the reinforcement means, i.e., for the expectations of the replacement matrix's diagonal elements, and prove their joint asymptotic normality. It is noteworthy that the estimators of the largest reinforcement mean are asymptotically independent of the estimators of the other smaller reinforcement means. Additionally, if a reinforcement mean is not the largest, the estimators of these smaller reinforcement means will also demonstrate asymptotic independence among themselves. Furthermore, we explore the parallels between the reinforced mechanisms in random reinforced urn models and multi-armed bandits, addressing hypothesis testing for expected payoffs in the latter context.

In this paper, we investigate the asymptotic behaviors of the extreme eigenvectors in a general spiked covariance matrix, where the dimension and sample size increase proportionally. We eliminate the restrictive assumption of the block diagonal structure in the population covariance matrix. Moreover, there is no requirement for the spiked eigenvalues and the 4th moment to be bounded. Specifically, we apply random matrix theory to derive the convergence and limiting distributions of certain projections of the extreme eigenvectors in a large sample covariance matrix within a generalized spiked population model. Furthermore, our techniques are robust and effective, even when spiked eigenvalues differ significantly in magnitude from nonspiked ones. Finally, we propose a powerful statistic for hypothesis testing for the eigenspaces of covariance matrices.

In this paper, we investigate hypothesis testing for the linear combination of mean vectors across multiple populations through the method of random integration. We have established the asymptotic distributions of the test statistics under both null and alternative hypotheses. Additionally, we provide a theoretical explanation for the special use of our test statistics in situations when the nonzero signal in the linear combination of the true mean vectors is weakly dense. Moreover, Monte-Carlo simulations are presented to evaluate the suggested test against existing high-dimensional tests. The findings from these simulations reveal that our test not only aligns with the performance of other tests in terms of size but also exhibits superior power.

In this paper, we propose a new modified likelihood ratio test (LRT) for simultaneously testing mean vectors and covariance matrices of two-sample populations in high-dimensional settings. By employing tools from Random Matrix Theory (RMT), we derive the limiting null distribution of the modified LRT for generally distributed populations. Furthermore, we compare the proposed test with existing tests using simulation results, demonstrating that the modified LRT exhibits favorable properties in terms of both size and power.

In this paper, we focus on the BDS test, which is a nonparametric test of independence. Specifically, the null hypothesis $H_{0}$ of it is that $\{u_{t}\}$ is i.i.d. (independent and identically distributed), where $\{u_{t}\}$ is a random sequence. The BDS test is widely used in economics and finance, but it has a weakness that cannot be ignored: over-rejecting $H_{0}$ even if the length $T$ of $\{u_{t}\}$ is as large as $(100,2000)$. To improve the over-rejection problem of BDS test, considering that the correlation integral is the foundation of BDS test, we not only accurately describe the expectation of the correlation integral under $H_{0}$, but also calculate all terms of the asymptotic variance of the correlation integral whose order is $O(T^{-1})$ and $O(T^{-2})$, which is essential to improve the finite sample performance of BDS test. Based on this, we propose a revised BDS (RBDS) test and prove its asymptotic normality under $H_{0}$. The RBDS test not only inherits all the advantages of the BDS test, but also effectively corrects the over-rejection problem of the BDS test, which can be fully confirmed by the simulation results we presented. Moreover, based on the simulation results, we find that similar to BDS test, RBDS test would also be affected by the parameter estimations of the ARCH-type model, resulting in size distortion, but this phenomenon can be alleviated by the logarithmic transformation preprocessing of the estimate residuals of the model. Besides, through some actual datasets that have been demonstrated to fit well with ARCH-type models, we also compared the performance of BDS test and RBDS test in evaluating the goodness-of-fit of the model in empirical problem, and the results reflect that, under the same condition, the performance of the RBDS test is more encouraging.

In this paper, we propose a novel approach to test the equality of high-dimensional mean vectors of several populations via the weighted $L_2$-norm. We establish the asymptotic normality of the test statistics under the null hypothesis. We also explain theoretically why our test statistics can be highly useful in weakly dense cases when the nonzero signal in mean vectors is present. Furthermore, we compare the proposed test with existing tests using simulation results, demonstrating that the weighted $L_2$-norm-based test statistic exhibits favorable properties in terms of both size and power.

In opportunistic cognitive radio networks, when the primary signal is very weak compared to the background noise, the secondary user requires long sensing time to achieve a reliable spectrum sensing performance, leading to little remaining time for the secondary transmission. To tackle this issue, we propose an active reconfigurable intelligent surface (RIS) assisted spectrum sensing system, where the received signal strength from the interested primary user can be enhanced and underlying interference within the background noise can be mitigated as well. In comparison with the passive RIS, the active RIS can not only adapt the phase shift of each reflecting element but also amplify the incident signals. Notably, we study the reflecting coefficient matrix (RCM) optimization problem to improve the detection probability given a maximum tolerable false alarm probability and limited sensing time. Then, we show that the formulated problem can be equivalently transformed to a weighted mean square error minimization problem using the principle of the well-known weighted minimum mean square error (WMMSE) algorithm, and an iterative optimization approach is proposed to obtain the optimal RCM. In addition, to fairly compare passive RIS and active RIS, we study the required power budget of the RIS to achieve a target detection probability under a special case where the direct links are neglected and the RIS-related channels are line-of-sight. Via extensive simulations, the effectiveness of the WMMSE-based RCM optimization approach is demonstrated. Furthermore, the results reveal that the active RIS can outperform the passive RIS when the underlying interference within the background noise is relatively weak, whereas the passive RIS performs better in strong interference scenarios because the same power budget can support a vast number of passive reflecting elements for interference mitigation.

Let $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $ where $ \bbX_n $ is a $ p \times n $ matrix with independent standardized random variables, $ \bbR_n $ is a $ p \times n $ non-random matrix, representing the information, and $ \bbT_{n} $ is a $ p \times p $ non-random nonnegative definite Hermitian matrix. Under some conditions on $ \bbR_n \bbR_n^* $ and $ \bbT_n $, it has been proved that for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all $ p $ sufficiently large. The purpose of this paper is to carry on with the study of the support of the limit spectral distribution, and we show that there is an exact separation phenomenon: with probability one, the proper number of eigenvalues lie on either side of these intervals.

An important issue in many multivariate regression problems is to eliminate candidate predictors with null predictor vectors. In large-dimensional (LD) setting where the numbers of responses and predictors are large, model selection encounters the scalability challenge. Knock-one-out (KOO) statistics hold promise to meet this challenge. In this paper, the almost sure limits and the central limit theorem of the KOO statistics are derived under the LD setting and mild distributional assumptions (finite fourth moments) of the errors. These theoretical results guarantee the strong consistency of a subset selection rule based on the KOO statistics with a general threshold. For enhancing the robustness of the selection rule, we also propose a bootstrap threshold for the KOO approach. Simulation results support our conclusions and demonstrate the selection probabilities by the KOO approach with the bootstrap threshold outperform the methods using Akaike information threshold, Bayesian information threshold and Mallow's C$_p$ threshold. We compare the proposed KOO approach with those based on information threshold to a chemometrics dataset and a yeast cell-cycle dataset, which suggests our proposed method identifies useful models.

Let $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $, where $ \bbX_n $ is a $ p \times n $ matrix with independent standardized random variables, $ \bbR_n $ is a $ p \times n $ non-random matrix and $ \bbT_{n} $ is a $ p \times p $ non-random, nonnegative definite Hermitian matrix. The matrix $\bbB_n$ is referred to as the information-plus-noise type matrix, where $\bbR_n$ contains the information and $\bbT^{1/2}_n \bbX_n$ is the noise matrix with the covariance matrix $\bbT_{n} $. It is known that, as $ n \to \infty $, if $ p/n $ converges to a positive number, the empirical spectral distribution of $ \bbB_n $ converges almost surely to a nonrandom limit, under some mild conditions. In this paper, we prove that, under certain conditions on the eigenvalues of $ \bbR_n $ and $ \bbT_n $, for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all $ n $ sufficiently large., Comment: 26 pages

In this paper, the fused graphical lasso (FGL) method is used to estimate multiple precision matrices from multiple populations simultaneously. The lasso penalty in the FGL model is a restraint on sparsity of precision matrices, and a moderate penalty on the two precision matrices from distinct groups restrains the similar structure across multiple groups. In high-dimensional settings, an oracle inequality is provided for FGL estimators, which is necessary to establish the central limit law. We not only focus on point estimation of a precision matrix, but also work on hypothesis testing for a linear combination of the entries of multiple precision matrices. Inspired by Jankova a and van de Geer [confidence intervals for high-dimensional inverse covariance estimation, Electron. J. Stat. 9(1) (2015) 1205-1229.], who investigated a de-biasing technology to obtain a new consistent estimator with known distribution for implementing the statistical inference, we extend the statistical inference problem to multiple populations, and propose the de-biasing FGL estimators. The corresponding asymptotic property of de-biasing FGL estimators is provided. A simulation study shows that the proposed test works well in high-dimensional situations.

In this paper, we derive the analytical behavior of the limiting spectral distribution of non-central covariance matrices of the "general information-plus-noise" type, as studied in [14]. Through the equation defining its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and we show the determination criterion for its support. We also extend the result in [14] to allow for all possible ratios of row to column of the underlying random matrix.

In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSSs) of a large-dimensional sample covariance matrix when the population covariance matrices are involved with diverging spikes. This constitutes a nontrivial extension of the Bai-Silverstein theorem (BST) (Ann Probab 32(1):553--605, 2004), a theorem that has strongly influenced the development of high-dimensional statistics, especially in the applications of random matrix theory to statistics. Recently, there has been a growing realization that the assumption of uniform boundedness of the population covariance matrices in the BST is not satisfied in some fields, such as economics, where the variances of principal components may diverge as the dimension tends to infinity. Therefore, in this paper, we aim to eliminate this obstacle to applications of the BST. Our new CLT accommodates spiked eigenvalues, which may either be bounded or tend to infinity. A distinguishing feature of our result is that the variance in the new CLT is related to both spiked eigenvalues and bulk eigenvalues, with dominance being determined by the divergence rate of the largest spiked eigenvalues. The new CLT for LSS is then applied to test the hypothesis that the population covariance matrix is the identity matrix or a generalized spiked model. The asymptotic distributions of the corrected likelihood ratio test statistic and the corrected Nagao's trace test statistic are derived under the alternative hypothesis. Moreover, we present power comparisons between these two LSSs and Roy's largest root test. In particular, we demonstrate that except for the case in which the number of spikes is equal to one, the LSSs could exhibit higher asymptotic power than Roy's largest root test., Comment: It is an update version of arXiv:2212.05896

Let $G$ be a finite Abelian group of order $d$. We consider an urn in which, initially, there are labeled balls that generate the group $G$. Choosing two balls from the urn with replacement, observe their labels, and perform a group multiplication on the respective group elements to obtain a group element. Then, we put a ball labeled with that resulting element into the urn. This model was formulated by P. Diaconis while studying a group theoretic algorithm called MeatAxe (Holt and Rees (1994)). Siegmund and Yakir (2004) partially investigated this model. In this paper, we further investigate and generalize this model. More specifically, we allow a random number of balls to be drawn from the urn at each stage in the Diaconis urn model. For such a case, we verify that the normalized urn composition converges almost surely to the uniform distribution on the group $G$. Moreover, we obtain the asymptotic joint distribution of the urn composition by using the martingale central limit theorem.

This paper proposes an improved linear discriminant analysis called spectrally-corrected and regularized LDA (SRLDA). This method integrates the design ideas of the sample spectrally-corrected covariance matrix and the regularized discriminant analysis. With the support of a large-dimensional random matrix analysis framework, it is proved that SRLDA has a linear classification global optimal solution under the spiked model assumption. According to simulation data analysis, the SRLDA classifier performs better than RLDA and ILDA and is closer to the theoretical classifier. Experiments on different data sets show that the SRLDA algorithm performs better in classification and dimensionality reduction than currently used tools.

In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix when the population covariance matrices are not uniformly bounded, which is a nontrivial extension of the Bai-Silverstein theorem (BST) (2004). The latter has strongly stimulated the development of high-dimensional statistics, especially the application of random matrix theory to statistics. However, the assumption of uniform boundedness of the population covariance matrices is found strongly limited to the applications of BST. The aim of this paper is to remove the blockages to the applications of BST. The new CLT, allows the spiked eigenvalues to exist and tend to infinity. It is interesting to note that the roles of either spiked eigenvalues or the bulk eigenvalues or both of the two are dominating in the CLT. Moreover, the results are checked by simulation studies with various population settings. The CLT for LSS is then applied for testing the hypothesis that a covariance matrix $ \bSi $ is equal to an identity matrix. For this, the asymptotic distributions for the corrected likelihood ratio test (LRT) and Nagao's trace test (NT) under alternative are derived, and we also propose the asymptotic power of LRT and NT under certain alternatives.

This paper aims to derive asymptotical distributions of the spiked eigenvalues of the large-dimensional spiked Fisher matrices without Gaussian assumption and the restrictive assumptions on covariance matrices. We first establish invariance principle for the spiked eigenvalues of the Fisher matrix. That is, we show that the limiting distributions of the spiked eigenvalues are invariant over a large class of population distributions satisfying certain conditions. Using the invariance principle, we further established a central limit theorem (CLT) for the spiked eigenvalues. As some interesting applications, we use the CLT to derive the power functions of Roy Maximum root test for linear hypothesis in linear models and the test in signal detection. We conduct some Monte Carlo simulation to compare the proposed test with existing ones., Comment: 19 pages; 3 figures. arXiv admin note: text overlap with arXiv:1904.09236

Let $ X_{n} $ be $ n\times N $ random complex matrices, $R_{n}$ and $T_{n}$ be non-random complex matrices with dimensions $n\times N$ and $n\times n$, respectively. We assume that the entries of $ X_{n} $ are independent and identically distributed, $ T_{n} $ are nonnegative definite Hermitian matrices and $T_{n}R_{n}R_{n}^{*}= R_{n}R_{n}^{*}T_{n} $. The general information-plus-noise type matrices are defined by $C_{n}=\frac{1}{N}T_{n}^{\frac{1}{2}} \left( R_{n} +X_{n}\right) \left(R_{n}+X_{n}\right)^{*}T_{n}^{\frac{1}{2}} $. In this paper, we establish the limiting spectral distribution of the large dimensional general information-plus-noise type matrices $C_{n}$. Specifically, we show that as $n$ and $N$ tend to infinity proportionally, the empirical distribution of the eigenvalues of $C_{n}$ converges weakly to a non-random probability distribution, which is characterized in terms of a system of equations of its Stieltjes transform.

In this paper, we investigate the asymptotic behavior of spiked eigenvalues of the noncentral Fisher matrix defined by ${\mathbf F}_p={\mathbf C}_n(\mathbf S_N)^{-1}$, where ${\mathbf C}_n$ is a noncentral sample covariance matrix defined by $(\mathbf \Xi+\mathbf X)(\mathbf \Xi+\mathbf X)^*/n$ and $\mathbf S_N={\mathbf Y}{\mathbf Y}^*/N$. The matrices $\mathbf X$ and $\mathbf Y$ are two independent {Gaussian} arrays, with respective $p\times n$ and $p\times N$ and the Gaussian entries of them are \textit {independent and identically distributed} (i.i.d.) with mean $0$ and variance $1$. When $p$, $n$, and $N$ grow to infinity proportionally, we establish a phase transition of the spiked eigenvalues of $\mathbf F_p$. Furthermore, we derive the \textit{central limiting theorem} (CLT) for the spiked eigenvalues of $\mathbf F_p$. As an accessory to the proof of the above results, the fluctuations of the spiked eigenvalues of ${\mathbf C}_n$ are studied, which should have its own interests. Besides, we develop the limits and CLT for the sample canonical correlation coefficients by the results of the spiked noncentral Fisher matrix and give three consistent estimators, including the population spiked eigenvalues and the population canonical correlation coefficients.

Large-dimensional random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into large dimensional systems. With the fact that we have entered an unprecedented era full of massive amounts of data and large complex systems, RMT is expected to play more important roles in the analysis and design of modern systems. In this paper, we review the key results of RMT and its applications in two emerging fields: wireless communications and deep learning. In wireless communications, we show that RMT can be exploited to design the spectrum sensing algorithms for cognitive radio systems and to perform the design and asymptotic analysis for large communication systems. In deep learning, RMT can be utilized to analyze the Hessian, input-output Jacobian and data covariance matrix of the deep neural networks, thereby to understand and improve the convergence and the learning speed of the neural networks. Finally, we highlight some challenges and opportunities in applying RMT to the practical large dimensional systems.

This paper presents a 4-channel beamforming TX implemented in 65nm CMOS. Each beamforming TX is comprised of a C-2C split-array multiphase switched-capacitor power amplifier (SAMP-SCPA). This is the first use of multiphase interpolation (MPI) for beam-steering. This technique is ideal for low-frequency beamforming and MIMO, as it does not require passive or LO based phase shifters. The SCPA is ideal to use as the core element since it can perform frequency translation, data conversion and drive an output at high power and efficiency in a compact die area. A prototype 4-element beamforming TX, occupying 2 mm X 2.5 mm, can achieve peak output power of 24.4 dBm with a peak system efficiency (SE) of 24%, while achieving < 1{\deg} phase resolution and <1 dB gain error. When transmitting a 15 MHz, 64 QAM long-term evolution (LTE) signal it outputs 18.4 dBm at 14% SE with a measured adjacent channel leakage ratio (ACLR) < -30 dBc and error vector magnitude (EVM) of 3.27 %-rms at 1.75 GHz. A synthesized beam pattern based on measured results from a single die achieves <0.32{\deg}-rms beam angle error and <0.15 dB rms beam amplitude error., Comment: 12 Pages, 27 Figures, 2 Tables. Expansion of paper 4.3 from ISSCC 2019

The goal of this study was to test the equality of two covariance matrices by using modified Pillai's trace statistics under a high-dimensional framework, i.e., the dimension and sample sizes go to infinity proportionally. In this paper, we introduce two modified Pillai's trace statistics and obtain their asymptotic distributions under the null hypothesis. The benefits of the proposed statistics include the following: (1) the sample size can be smaller than the dimensions; (2) the limiting distributions of the proposed statistics are universal; and (3) we do not restrict the structure of the population covariance matrices. The theoretical results are established under mild and practical assumptions, and their properties are demonstrated numerically by simulations and a real data analysis., Comment: 31 pages-to appear in Journal of Statistical Planning and Inference

Spectral clustering is one of the most popular algorithms for community detection in network analysis. Based on this rationale, in this paper we give the convergence rate of eigenvectors for the adjacency matrix in the $l_\infty$ norm, under the stochastic block model (BM) and degree corrected stochastic block model (DCBM), adding some mild and rational conditions. We also extend this result to a more general model, presented based on the DCBM such that the value of random variables in the adjacency matrix is not 0 or 1, but an arbitrary real number. During the process of proving the above conclusion, we obtain the relationship of the eigenvalues in the adjacency matrix and the corresponding `population' matrix, which vary in dimension from the community-wise edge probability matrix. Using that result, we can give an estimate of the number of the communities in a known set of network data. Meanwhile we proved the consistency of the estimator. Furthermore, according to the derivation of proof for the convergence of eigenvectors, we propose a new approach to community detection -- Spectral Clustering based on Difference of Ratios of Eigenvectors (SCDRE). Our simulation experiments demonstrate the superiority of our method in community detection., Comment: 28 pages, 2 figures

The universality for the local spiked eigenvalues is a powerful tool to deal with the problems of the asymptotic law for the bulks of spiked eigenvalues of high-dimensional generalized Fisher matrices. In this paper, we focus on a more generalized spiked Fisher matrix, where $\Sigma_1\Sigma_2^{-1}$ is free of the restriction of diagonal independence, and both of the spiked eigenvalues and the population 4th moments are not necessary required to be bounded. By reducing the matching four moments constraint to a tail probability, we propose a Generalized Four Moment Theorem (G4MT) for the bulks of spiked eigenvalues of high-dimensional generalized Fisher matrices, which shows that the limiting distribution of the spiked eigenvalues of a generalized spiked Fisher matrix is independent of the actual distributions of the samples provided to satisfy the our relaxed assumptions. Furthermore, as an illustration, we also apply the G4MT to the Central Limit Theorem for the spiked eigenvalues of generalized spiked Fisher matrix, which removes the strict condition of the diagonal block independence given in Wang and Yao (2017) and extends their result to a wider usage without the requirements of the bounded 4th moments and the diagonal block independent structure, meeting the actual cases better., Comment: 29pages, 4 figures

Variable selection is essential for improving inference and interpretation in multivariate linear regression. Although a number of alternative regressor selection criteria have been suggested, the most prominent and widely used are the Akaike information criterion (AIC), Bayesian information criterion (BIC), Mallow's $C_p$, and their modifications. However, for high-dimensional data, experience has shown that the performance of these classical criteria is not always satisfactory. In the present article, we begin by presenting the necessary and sufficient conditions (NSC) for the strong consistency of the high-dimensional AIC, BIC, and $C_p$, based on which we can identify some reasons for their poor performance. Specifically, we show that under certain mild high-dimensional conditions, if the BIC is strongly consistent, then the AIC is strongly consistent, but not vice versa. This result contradicts the classical understanding. In addition, we consider some NSC for the strong consistency of the high-dimensional kick-one-out (KOO) methods introduced by Zhao et al. (1986) and Nishii et al. (1988). Furthermore, we propose two general methods based on the KOO methods and prove their strong consistency. The proposed general methods remove the penalties while simultaneously reducing the conditions for the dimensions and sizes of the regressors. A simulation study supports our consistency conclusions and shows that the convergence rates of the two proposed general KOO methods are much faster than those of the original methods.

We consider a more generalized spiked covariance matrix $\Sigma$, which is a general non-definite matrix with the spiked eigenvalues scattered into a few bulks and the largest ones allowed to tend to infinity. By relaxing the matching of the 4th moment to a tail probability decay, a {\it Generalized Four Moment Theorem} (G4MT) is proposed to show the universality of the asymptotic law for the local spectral statistics of generalized spiked covariance matrices, which implies the limiting distribution of the spiked eigenvalues of the generalized spiked covariance matrix is independent of the actual distributions of the samples satisfying our relaxed assumptions. Moreover, by applying it to the Central Limit Theorem (CLT) for the spiked eigenvalues of the generalized spiked covariance matrix, we also extend the result of Bai and Yao (2012) to a general form of the population covariance matrix, where the 4th moment is not necessarily required to exist and the spiked eigenvalues are allowed to be dependent on the non-spiked ones, thus meeting the actual cases better., Comment: 48 pages, 4 figures,5 tables

As one fundamental property of light, the orbital angular momentum (OAM) of photon has elicited widespread interest. Here, we theoretically demonstrate that the OAM conversion of light without any spin state can occur in homogeneous and isotropic medium when a specially tailored locally linearly polarized (STLLP) beam is strongly focused by a high numerical aperture (NA) objective lens. Through a high NA objective lens, the STLLP beams can generate identical twin foci with tunable distance between them controlled by input state of polarization. Such process admits partial OAM conversion from linear state to conjugate OAM states, giving rise to helical phases with opposite directions for each focus of the longitudinal component in the focal field.

This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form $\mathbf{B}_n=n^{-1}\sum_{j=1}^{n}\mathbf{Q}\mathbf{x}_j\mathbf{x}_j^{*}\mathbf{Q}^{*}$ where $\mathbf{Q}$ is a nonrandom matrix of dimension $p\times k$, and $\{\mathbf{x}_j\}$ is a sequence of independent $k$-dimensional random vector with independent entries, under the assumption that $p/n\to y>0$. A key novelty here is that the dimension $k\ge p$ can be arbitrary, possibly infinity. This new model of sample covariance matrices $\mathbf{B}_n$ 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 (2004). Applications of this new CLT are proposed for testing the structure of a high-dimensional covariance matrix. The derived tests are then used to analyse a large fMRI data set regarding its temporary correlation structure., Comment: 46 pages, 1 figure. This version of the paper is written on July 13, 2016

Magnetoencephalography (MEG) is an advanced imaging technique used to measure the magnetic fields outside the human head produced by the electrical activity inside the brain. Various source localization methods in MEG require the knowledge of the underlying active sources, which are identified by a priori. Common methods used to estimate the number of sources include principal component analysis or information criterion methods, both of which make use of the eigenvalue distribution of the data, thus avoiding solving the time-consuming inverse problem. Unfortunately, all these methods are very sensitive to the signal-to-noise ratio (SNR), as examining the sample extreme eigenvalues does not necessarily reflect the perturbation of the population ones. To uncover the unknown sources from the very noisy MEG data, we introduce a framework, referred to as the intrinsic dimensionality (ID) of the optimal transformation for the SNR rescaling functional. It is defined as the number of the spiked population eigenvalues of the associated transformed data matrix. It is shown that the ID yields a more reasonable estimate for the number of sources than its sample counterparts, especially when the SNR is small. By means of examples, we illustrate that the new method is able to capture the number of signal sources in MEG that can escape PCA or other information criterion based methods., Comment: 38 pages, 8 figures, 4 tables

This paper considers the optimal modification of the likelihood ratio test (LRT) for the equality of two high-dimensional covariance matrices. The classical LRT is not well defined when the dimensions are larger than or equal to one of the sample sizes. In this paper, an optimally modified test that works well in cases where the dimensions may be larger than the sample sizes is proposed. In addition, the test is established under the weakest conditions on the moments and the dimensions of the samples. We also present weakly consistent estimators of the fourth moments, which are necessary for the proposed test, when they are not equal to 3. From the simulation results and real data analysis, we find that the performances of the proposed statistics are robust against affine transformations.

The multivariate nonlinear Granger causality developed by Bai et al. (2010) plays an important role in detecting the dynamic interrelationships between two groups of variables. Following the idea of Hiemstra-Jones (HJ) test proposed by Hiemstra and Jones (1994), they attempt to establish a central limit theorem (CLT) of their test statistic by applying the asymptotical property of multivariate $U$-statistic. However, Bai et al. (2016) revisit the HJ test and find that the test statistic given by HJ is NOT a function of $U$-statistics which implies that the CLT neither proposed by Hiemstra and Jones (1994) nor the one extended by Bai et al. (2010) is valid for statistical inference. In this paper, we re-estimate the probabilities and reestablish the CLT of the new test statistic. Numerical simulation shows that our new estimates are consistent and our new test performs decent size and power., Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1701.03992

The famous Hiemstra-Jones (HJ) test developed by Hiemstra and Jones (1994) plays a significant role in studying nonlinear causality. Over the last two decades, there have been numerous applications and theoretical extensions based on this pioneering work. However, several works note that counterintuitive results are obtained from the HJ test, and some researchers find that the HJ test is seriously over-rejecting in simulation studies. In this paper, we reinvestigate HJ's creative 1994 work and find that their proposed estimators of the probabilities over different time intervals were not consistent with the target ones proposed in their criterion. To test HJ's novel hypothesis on Granger causality, we propose new estimators of the probabilities defined in their paper and reestablish the asymptotic properties to induce new tests similar to those of HJ. Some simulations will also be presented to support our findings.

In this paper, we adopt the eigenvector empirical spectral distribution (VESD) to investigate the limiting behavior of eigenvectors of a large dimensional Wigner matrix W_n. In particular, we derive the optimal bound for the rate of convergence of the expected VESD of W_n to the semicircle law, which is of order O(n^{-1/2}) under the assumption of having finite 10th moment. We further show that the convergence rates in probability and almost surely of the VESD are O(n^{-1/4}) and O(n^{-1/6}), respectively, under finite 8th moment condition. Numerical studies demonstrate that the convergence rate does not depend on the choice of unit vector involved in the VESD function, and the best possible bound for the rate of convergence of the VESD is of order O(n^{-1/2}).

This paper is to prove the asymptotic normality of a statistic for detecting the existence of heteroscedasticity for linear regression models without assuming randomness of covariates when the sample size $n$ tends to infinity and the number of covariates $p$ is either fixed or tends to infinity. Moreover our approach indicates that its asymptotic normality holds even without homoscedasticity., Comment: 19 pages

In this paper, we will introduce the so called naive tests and give a brief review on the newly development. Naive testing methods are easy to understand and performs robust especially when the dimension is large. In this paper, we mainly focus on reviewing some naive testing methods for the mean vectors and covariance matrices of high dimensional populations and believe this naive test idea can be wildly used in many other testing problems.

In this article, we focus on the problem of testing the equality of several high dimensional mean vectors with unequal covariance matrices. This is one of the most important problem in multivariate statistical analysis and there have been various tests proposed in the literature. Motivated by \citet{BaiS96E} and \cite{ChenQ10T}, a test statistic is introduced and the asymptomatic distributions under the null hypothesis as well as the alternative hypothesis are given. In addition, it is compared with a test statistic recently proposed by \cite{SrivastavaK13Ta}. It is shown that our test statistic performs much better especially in the large dimensional case.

Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two multivariate population covariance matrices, or testing the independence between sub-groups of a multivariate random vector. This paper is concerned with the properties of a large-dimensional Fisher matrix when the dimension of the population is proportionally large compared to the sample size. Most of existing works on Fisher matrices deal with a particular Fisher matrix where populations have i.i.d components so that the population covariance matrices are all identity. In this paper, we consider general Fisher matrices with arbitrary population covariance matrices. The first main result of the paper establishes the limiting distribution of the eigenvalues of a Fisher matrix while in a second main result, we provide a central limit theorem for a wide class of functionals of its eigenvalues. Some applications of these results are also proposed for testing hypotheses on high-dimensional covariance matrices.

In Jin et al. (2014), the limiting spectral distribution (LSD) of a symmetrized auto-cross covariance matrix is derived using matrix manipulation, with finite $(2+\delta)$-th moment assumption. Here we give an alternative method using a result in Bai and Silverstein (2010), in which a weaker condition of finite 2nd moment is assumed.

In this paper, we study the convergence rates of empirical spectral distribution of large dimensional quaternion sample covariance matrix. Assume that the entries of $\mathbf X_n$ ($p\times n$) are independent quaternion random variables with mean zero, variance 1 and uniformly bounded sixth moments. Denote $\mathbf S_n=\frac{1}{n}\mathbf X_n\mathbf X_n^*$. Using Bai inequality, we prove that the expected empirical spectral distribution (ESD) converges to the limiting Mar${\rm \check{c}}$enko-Pastur distribution with the ratio of the dimension to sample size $y_p=p/n$ at a rate of $O\left(n^{-1/2}a_n^{-3/4}\right)$ when $a_n>n^{-2/5}$ or $O\left(n^{-1/5}\right)$ when $a_n\le n^{-2/5}$, where $a_n=(1-\sqrt{y_p})^2$ is the lower bound for the M-P law. Moreover, the rates for both the convergence in probability and the almost sure convergence are also established. The weak convergence rate of the ESD is $O\left(n^{-2/5}a_n^{-2/5}\right)$ when $a_n>n^{-2/5}$ or $O\left(n^{-1/5}\right)$ when $a_n\le n^{-2/5}$. The strong convergence rate of the ESD is $O\left(n^{-2/5+\eta}a_n^{-2/5}\right)$ when $a_n>n^{-2/5}$ or $O\left(n^{-1/5}\right)$ when $a_n\le n^{-2/5}$ for any $\eta>0$., Comment: 25

In this paper, we shall investigate the almost sure limits of the largest and smallest eigenvalues of a quaternion sample covariance matrix. Suppose that $\mathbf X_n$ is a $p\times n$ matrix whose elements are independent quaternion variables with mean zero, variance 1 and uniformly bounded fourth moments. Denote $\mathbf S_n=\frac{1}{n}\mathbf X_n\mathbf X_n^*$. In this paper, we shall show that $s_{\max}\left(\mathbf S_n\right)=s_{p}\left(\mathbf S_n\right)\to\left(1+\sqrt y\right)^2, a.s.$ and $s_{\min}\left(\mathbf S_n\right)\to\left(1-\sqrt y\right)^2,a.s.$ as $n\to\infty$, where $y=\lim p/n$, $s_1\left(\mathbf S_n\right)\le\cdots\le s_{p}\left(\mathbf S_n\right)$ are the eigenvalues of $\mathbf{S}_n$, $s_{\min}\left(\mathbf S_n\right)=s_{p-n+1}\left(\mathbf S_n\right)$ when $p>n$ and $s_{\min}\left(\mathbf S_n\right)=s_1\left(\mathbf S_n\right)$ when $p\le n$. We also prove that the set of conditions are necessary for $s_{\max}\left(\mathbf S_n\right)\to\left(1+\sqrt y\right)^2, a.s.$ when the entries of $\mathbf {X}_n$ are i. i. d., Comment: 21 pages

In this paper, convergence rates of the spectral distributions of quaternion self-dual Hermitian matrices are investigated. We show that under conditions of finite 6th moments, the expected spectral distribution of a large quaternion self-dual Hermitian matrix converges to the semicircular law in a rate of $O(n^{-1/2})$ and the spectral distribution itself converges to the semicircular law in rates $O_p(n^{-2/5})$ and $O_{a.s.}(n^{-2/5+\eta})$. Those results include GSE as a special case.

Since E.P.Wigner (1958) established his famous semicircle law, lots of attention has been paid by physicists, probabilists and statisticians to study the asymptotic properties of the largest eigenvalues for random matrices. Bai and Yin (1988) obtained the necessary and sufficient conditions for the strong convergence of the extreme eigenvalues of a Wigner matrix. In this paper, we consider the case of quaternion self-dual Hermitian matrices. We prove the necessary and sufficient conditions for the strong convergence of extreme eigenvalues of quaternion self-dual Hermitian matrices corresponding to the Wigner case., Comment: 16 pages, 5 figures

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of sample covariance matrix and the Mar\v{c}enko-Pastur distribution function is of order $O(N^{-1/2})$. Given that data dimension $n$ to sample size $N$ ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed $\eta>0$, the convergence rates of VESD are $O(N^{-1/4})$ in probability and $O(N^{-1/4+\eta})$ almost surely, requiring finite 8th moment of the underlying distribution., Comment: Published in at http://dx.doi.org/10.1214/13-AOS1154 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper we establish the limit of the empirical spectral distribution of quaternion sample covariance matrices. Suppose $\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}$ is a quaternion random matrix. For each $n$, the entries $\{x_{ij}^{(n)}\}$ are independent random quaternion variables with a common mean $\mu$ and variance $\sigma^2>0$. It is shown that the empirical spectral distribution of the quaternion sample covariance matrix $\mathbf S_n=n^{-1}\mathbf X_n\mathbf X_n^*$ converges to the M-P law as $p\to\infty$, $n\to\infty$ and $p/n\to y\in(0,+\infty)$., Comment: 20 pages

Skorokhod's representation theorem states that if on a Polish space, there is defined a weakly convergent sequence of probability measures $\mu_n\stackrel{w}\to\mu_0,$ as $n\to \infty$, then there exist a probability space $(\Omega, \mathscr F, P)$ and a sequence of random elements $X_n$ such that $X_n\to X$ almost surely and $X_n$ has the distribution function $\mu_n$, $n=0,1,2,\cdots$. In this paper, we shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces $S_n$, a sequence of probability measures $\mu_n$ and a sequence of measurable mappings $\varphi_n$ such that $\mu_n\varphi_n^{-1}\stackrel {w}\to\mu_0$, then there exist a probability space $(\Omega,\mathscr F,P)$ and $S_n$-valued random elements $X_n$ defined on $\Omega$, with distribution $\mu_n$ and such that $\varphi_n(X_n)\to X_0$ almost surely. In addition, we present several applications of our result including some results in random matrix theory, while the original Skorokhod representation theorem is not applicable., Comment: 11 pages

It is well known that Gaussian symplectic ensemble (GSE) is defined on the space of $n\times n$ quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices. As a natural idea we want to get more universal results by removing the Gaussian condition. For the first step, in this paper we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well., Comment: 20 pages

This paper proposes a CLT for linear spectral statistics of random matrix $S^{-1}T$ for a general non-negative definite and {\bf non-random} Hermitian matrix $T$.

Sample covariance matrix and multivariate $F$-matrix play important roles in multivariate statistical analysis. The central limit theorems {\sl (CLT)} of linear spectral statistics associated with these matrices were established in Bai and Silverstein (2004) and Zheng (2012) which received considerable attentions and have been applied to solve many large dimensional statistical problems. However, the sample covariance matrices used in these papers are not centralized and there exist some questions about CLT's defined by the centralized sample covariance matrices. In this note, we shall provide some short complements on the CLT's in Bai and Silverstein (2004) and Zheng (2012), and show that the results in these two papers remain valid for the centralized sample covariance matrices, provided that the ratios of dimension $p$ to sample sizes $(n,n_1,n_2)$ are redefined as $p/(n-1)$ and $p/(n_i-1)$, $i=1,2$, respectively.

This paper introduces a new method to estimate the spectral distribution of a population covariance matrix from high-dimensional data. The method is founded on a meaningful generalization of the seminal Marcenko-Pastur equation, originally defined in the complex plan, to the real line. Beyond its easy implementation and the established asymptotic consistency, the new estimator outperforms two existing estimators from the literature in almost all the situations tested in a simulation experiment. An application to the analysis of the correlation matrix of S&P stocks data is also given., Comment: 16 pages, 4 figures