Random Matrices: Theory and Applications

In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta $F$-matrix and extend the distributions of a nonsingular beta $F$-matrix to the singular case. We also give the joint density of eigenvalues and the exact distribution of the largest eigenvalue in terms of heterogeneous hypergeometric functions.

We study the high-dimensional asymptotic regimes of correlated Wishart matrices $d^{-1}{𝒴}{{𝒴}}^T$, where ${𝒴}$ is a $n\times d$ Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both $n$ and $d$ grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes close in Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the second regime, a non-central convergence happens, and the law of the normalized Wishart matrices becomes close in Wasserstein distance to that of the so-called Rosenblatt–Wishart matrix recently introduced by Nourdin and Zheng. We then proceed to show that the convergences stated above also hold in a functional setting, namely as weak convergence in $C([a,b];M_n(ℝ))$. As an application of our main result (in the central convergence regime), we show that it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. Our findings complement and extend a rich collection of results on the study of the fluctuations of Gaussian Wishart matrices, and we provide explicit examples based on Gaussian entries given by normalized increments of a bi-fractional or a sub-fractional Brownian motion.

Consider two random vectors $C_1^{1/2}x\in {ℝ}^p$ and $C_2^{1/2}y\in {ℝ}^q$, where the entries of $x$ and $y$ are i.i.d. random variables with mean zero and variance one, and $C_1$ and $C_2$ are respectively, $p\times p$ and $q\times q$ deterministic population covariance matrices. With $n$ independent samples of $(C_1^{1/2}x,C_2^{1/2}y)$, we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional setting with $p/n\to c_1\in (0,1)$ and $q/n\to c_2\in (0,1-c_1)$ as $n\to \infty$, we prove that the largest sample canonical correlation coefficient converges to the Tracy–Widom distribution as long as we have ${lim}_{s\to \infty }{s}^4\mathbb{P}(|x_{ij}|\ge s)=0$ and ${lim}_{s\to \infty }{s}^4\mathbb{P}(|y_{ij}|\ge s)=0$, which we believe to be a sharp moment condition. This extends the result in [19], which established the Tracy–Widom limit under the assumption that all moments exist for the entries of $x$ and $y$. Our proof is based on a new linearization method, which reduces the problem to the study of a $(p+q+2n)\times (p+q+2n)$ random matrix $H$. In particular, we shall prove an optimal local law on its inverse $G:=H^{-1}$, called resolvent. This local law is the main tool for both the proof of the Tracy–Widom law in this paper, and the study in [26, 27] on the canonical correlation coefficients of high-dimensional random vectors with finite rank correlations.

Optimal portfolio selection problems are determined by the (unknown) parameters of the data generating process. If an investor wants to realize the position suggested by the optimal portfolios, he/she needs to estimate the unknown parameters and to account for the parameter uncertainty in the decision process. Most often, the parameters of interest are the population mean vector and the population covariance matrix of the asset return distribution. In this paper, we characterize the exact sampling distribution of the estimated optimal portfolio weights and their characteristics. This is done by deriving their sampling distribution by its stochastic representation. This approach possesses several advantages, e.g. (i) it determines the sampling distribution of the estimated optimal portfolio weights by expressions, which could be used to draw samples from this distribution efficiently; (ii) the application of the derived stochastic representation provides an easy way to obtain the asymptotic approximation of the sampling distribution. The later property is used to show that the high-dimensional asymptotic distribution of optimal portfolio weights is a multivariate normal and to determine its parameters. Moreover, a consistent estimator of optimal portfolio weights and their characteristics is derived under the high-dimensional settings. Via an extensive simulation study, we investigate the finite-sample performance of the derived asymptotic approximation and study its robustness to the violation of the model assumptions used in the derivation of the theoretical results.

Complex Wishart matrices are a class of random matrices with numerous emerging applications. In particular, the statistical characterization of such class of random matrices is essential for solving problems in various fields, including statistics, finance, physics and engineering. This paper establishes a new way to solve such problems based on the statistical moments and correlation of the minors. The first two minors’ moments and correlation for noncentral complex Wishart matrices are derived as a function of size and degrees of freedom, as well as of covariance and noncentrality matrices. As a case study, the findings are applied to the statistical characterization of the capacity of wireless multiple-input–multiple-output systems.

With nonignorable dropouts and outliers, we propose robust statistical inference and variable selection methods for linear quantile regression models based on composite quantile regression and empirical likelihood (EL) that accommodate both the within-subject correlations and nonignorable dropouts. The purpose of our study is threefold. First, we apply the generalized method of moments to estimate the parameters in the nonignorable dropout propensity based on an instrument. Subsequently, the inverse probability weighting and kernel smoothing approaches are applied to obtain the smoothed and bias-corrected generalized estimating equations. Second, we borrow the idea of quadratic inference function to construct the improved EL procedure for nonignorable dropouts. The asymptotic properties of the proposed estimators and their confidence regions are derived. Third, the penalized EL method and algorithm for variable selection are investigated. The finite-sample performance of the proposed estimators is studied through simulation, and an application to HIV-CD4 data set is also presented.

In this paper, a ridgelized Hotelling’s $T^2$ test is developed for a hypothesis on a large-dimensional mean vector under certain moment conditions. It generalizes the main result of Chen et al. [A regularized Hotelling’s $t^2$ test for pathway analysis in proteomic studies, J. Am. Stat. Assoc. 106(496) (2011) 1345–1360.] by relaxing their Gaussian assumption. This is achieved by establishing an exact four-moment theorem that is a simplified version of Tao and Vu’s [Random matrices: universality of local statistics of eigenvalues, Ann. Probab. 40(3) (2012) 1285–1315] work. Simulation results demonstrate the superiority of the proposed test over the traditional Hotelling’s $T^2$ test and its several extensions in high-dimensional situations.

Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence.

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight