Random Matrices: Theory and Applications

This paper considers testing the covariance matrices structure based on Wald’s score test in large-dimensional setting. The tests for identity and sphericity of large-dimensional covariance matrices are reviewed by the generalized CLT for the linear spectral statistics of large-dimensional sample covariance matrices from [D. D. Jiang, Tests for large-dimensional covariance structure based on Rao’s score test, J. Multivariate Anal. 152 (2016) 28–39]. The proposed test can be applicable for large-dimensional non-Gaussian variables in a wider range. Furthermore, the simulation study is conducted to compare the proposed test with other large-dimensional covariance matrix tests. As seen from the simulation results, our proposed test is feasible for large-dimensional data without restriction of population distribution and provides the accurate and steady empirical sizes, which are almost around the nominal size.

This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues’ density, and write the universal limit in function of the Kontsevich matrix model (“matrix Airy function”, as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral (“matrix Bessel function”) known in the literature as generalized Kontsevich matrix model.

We consider four specific $n\times n$ sparse patterned random matrices, namely the Symmetric Circulant, Reverse Circulant, Toeplitz and the Hankel matrices. The entries are assumed to be Bernoulli with success probability $p_n$ such that $n{p}_n\to \xi$ with $0<\xi <\infty$. We use the moment approach to show that the expected empirical spectral distribution (EESD) converges weakly for all these sparse matrices. Unlike the Sparse Wigner matrices, here the random empirical spectral distribution (ESD) converges weakly to a random distribution. This weak convergence is only in the distribution sense. We give explicit description of the random limits of the ESD for Reverse Circulant and Circulant matrices. As in the non-sparse case, explicit description of the limits appears to be difficult to obtain in the Toeplitz and Hankel cases. We provide some properties of these limits. We then study the behavior of the largest eigenvalue of these matrices. We prove that for the Reverse Circulant and Symmetric Circulant matrices the limit distribution of the largest eigenvalue is a multiple of the Poisson. For Toeplitz and Hankel matrices we show that the non-degenerate limit distribution exists, but again it does not seem to be easy to obtain any explicit description.

We consider random $n\times n$ matrices of the form $Y_n=\frac{1}{\sqrt{d}}{A}_n\circ X_n$, where $A_n$ is the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with $d=\lfloor pn\rfloor$ for some fixed $p\in (0,1)$, and $X_n$ is an $n\times n$ matrix of i.i.d. centered random variables with unit variance and finite $(4+\eta )$th moment (here ∘ denotes the matrix Hadamard product). We show that as $n\to \infty$, the empirical spectral distribution of $Y_n$ converges weakly in probability to the normalized Lebesgue measure on the unit disk.