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 unpre
cedented 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.
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 med
ium 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 $mathbf{B}_n=n^{-1}sum_{j=1}^{n}mathbf{Q}mathbf{x}_jmathbf{x}_j^{*}mathbf{Q}^{*}$ where $mathbf{Q}$ is a nonra
ndom matrix of dimension $ptimes k$, and ${mathbf{x}_j}$ is a sequence of independent $k$-dimensional random vector with independent entries, under the assumption that $p/nto y>0$. A key novelty here is that the dimension $kge p$ 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.
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 covar
iates $p$ is either fixed or tends to infinity. Moreover our approach indicates that its asymptotic normality holds even without homoscedasticity.
Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of
large numbers. A result similar to Bahadurs representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that [sqrt{n}Biggl(frac{1}{n}sum_{i=1}^nphibigl(X_{n:i}^{(1)},...,X_{n:i}^{(d)}bigr)-bar{gamma}Biggr)=frac{1}{sqrt{n}}sum_{i=1}^nZ_{n,i}+mathrm{o}_P(1)] as $nrightarrowinfty$, where $bar{gamma}$ is a constant and $Z_{n,i}$ are i.i.d. random variables for each $n$. This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations.
