Random Matrices: Theory and Applications

In [Spectrum of SYK model, preprint (2018), arXiv:1801.10073], we proved the almost sure convergence of eigenvalues of the SYK model, which can be viewed as a type of law of large numbers in probability theory; in [Spectrum of SYK model II: Central limit theorem, preprint (2018), arXiv:1806.05714], we proved that the linear statistic of eigenvalues satisfies the central limit theorem. In this paper, we continue to study another important theorem in probability theory — the concentration of measure theorem, especially for the Gaussian SYK model. We will prove a large deviation principle (LDP) for the normalized empirical measure of eigenvalues when $q_n=2$, in which case the eigenvalues can be expressed in terms of these of Gaussian random antisymmetric matrices. Such LDP result has its own independent interest in random matrix theory. For general $q_n\ge 3$, we cannot prove the LDP, we will prove a concentration of measure theorem by estimating the Lipschitz norm of the Gaussian SYK model.

Let $A_n^{(i)}$, $1\le i\le k$, be $k$ probabilistically independent matrices of order $n_i\times n_{i+1}$ (with $n_1=n_{k+1}$) which are the left-uppermost blocks of $n\times n$ Haar unitary matrices. Suppose that $\frac{n}{n_i}\to {\alpha }_i$ as $n\to \infty$, with $1<{\alpha }_i<\infty$. Using free probability and Brown measure techniques, we find the limiting spectral distribution of $A_n^{(1)}\cdots A_n^{(k)}$.

Let ${\lambda }_{max},{\lambda }_{min}$ be respectively the largest and smallest eigenvalues of beta-Laguerre ensembles with parameters $(n,p,\beta )$. For fixed $\beta >0$, under the condition that $p$ is much larger than $n{(logn)}^2$, we obtain the full moderate deviation principles for ${\lambda }_{max}$ and ${\lambda }_{min}$ by using the asymptotic expansion technique. Interestingly, under this regime, our results show that asymptotically the exponential tails of the extreme eigenvalues are Gaussian-type distribution tail rather than the Tracy–Widom-type distribution tail.

The monitoring of high-dimensional data streams has become increasingly important for real-time detection of abnormal activities in many statistical process control (SPC) applications. Although the multivariate SPC has been extensively studied in the literature, the challenges associated with designing a practical monitoring scheme for high-dimensional processes when between-streams correlation exists are yet to be addressed well. Classical $T^2$-test-based schemes do not work well because the contamination bias in estimating the covariance matrix grows rapidly with the increase of dimension. We propose a test statistic which is based on the “divide-and-conquer” strategy, and integrate this statistic into the multivariate exponentially weighted moving average charting scheme for Phase II process monitoring. The key idea is to calculate the $T^2$ statistics on low-dimensional sub-vectors and to combine them together. The proposed procedure is essentially distribution-free and computation efficient. The control limit is obtained through the asymptotic distribution of the test statistic under some mild conditions on the dependence structure of stream observations. Our asymptotic results also shed light on quantifying the size of a reference sample required. Both theoretical analysis and numerical results show that the proposed method is able to control the false alarm rate and deliver robust change detection.

In this paper, we study the strong convergence of empirical spectral distribution (ESD) of the large quaternion sample covariance matrices and correlation matrices when the ratio of the population dimension $p$ to sample size $n$ tends to zero. We prove that the ESD of renormalized quaternion sample covariance matrices converges almost surely to the semicircle law.

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators to our problem, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ and show that they satisfy the coupled Riccati equations, from which we find that $R_n(t)$ satisfies the Painlevé V equation. Furthermore, we prove that ${\sigma }_n(t)$, a quantity related to the logarithmic derivative of the Hankel determinant, satisfies both the continuous and discrete Jimbo–Miwa–Okamoto $\sigma$-form of the Painlevé V. In the end, by using Dyson’s Coulomb fluid approach, we consider the large $n$ asymptotic behavior of our problem at the soft edge, which gives rise to the Painlevé XXXIV equation.

We consider a panel data partially linear single-index models (PDPLSIM) with errors correlated in space and time. A serially correlated error structure is adopted for the correlation in time. We propose using a semiparametric minimum average variance estimation (SMAVE) to obtain estimators for both the parameters and unknown link function. We not only establish an asymptotically normal distribution for the estimators of the parameters in the single index and the linear component of the model, but also obtain an asymptotically normal distribution for the nonparametric local linear estimator of the unknown link function. Then, a fitting of spatial and time-wise correlation structures is investigated. Based on the estimators, we propose a generalized F-type test method to deal with testing problems of index parameters of PDPLSIM with errors correlated in space and time. It is shown that under the null hypothesis, the proposed test statistic follows asymptotically a ${\chi }^2$-distribution with the scale constant and degrees of freedom being independent of nuisance parameters or functions. Simulated studies and real data examples have been used to illustrate our proposed methodology.