Random Matrices: Theory and Applications

In this paper, we propose time-varying coefficient autoregressive models (TVARMs) of $p$th order as a solution for capturing the nonlinear structure of multiple explanatory variables. The stationary $\alpha$-mixing sequence of autoregressive variables is considered. We apply empirical likelihood approach to investigate confidence regions of the autoregressive coefficients of the TVARMs, and establish the nonparametric version of Wilks’ theorem of the proposed empirical log-likelihood ratio. Further, the maximum empirical likelihood estimator of the autoregressive coefficients and its asymptotic distribution are also obtained. Besides, to determine the order of autoregression, penalized empirical likelihood approach is developed with the help of smoothly clipped absolute deviation penalty and the resultant penalized estimators have the oracle property. Simulation studies and an analysis of monthly Chinese price index dataset are used to demonstrate the effectiveness of our proposed methods.

Building on the work of Arizmendi and Celestino (2021), we derive the ∗-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers $\alpha$ and $\beta$, we derive explicitly the ∗-distribution of $p_{\alpha ,\beta }=\alpha ab+\beta ba$ whenever $a$ and $b$ are monotone or infinitesimally monotone independent elements. This encompasses both cases of the commutator and anti-commutator. This approach can be pushed to study more general polynomials. As applications, we derive the limiting distribution with respect to the partial trace of polynomials in a certain class of random matrices.

We study analytically the Chalker–Mehlig mean diagonal overlap ${𝒪}(z)$ between left and right eigenvectors associated with a complex eigenvalue $z$ of $N\times N$ matrices in the real Ginibre ensemble (GinOE). We first derive a general finite $N$ expression for the mean overlap and then investigate several scaling regimes in the limit $N\to \infty$. While in the generic spectral bulk and edge of the GinOE the limiting expressions for ${𝒪}(z)$ are found to coincide with the known results for the complex Ginibre ensemble (GinUE), in the region of eigenvalue depletion close to the real axis the asymptotic for the GinOE is considerably different. We also study numerically the distribution of diagonal overlaps and conjecture that it is the same in the bulk and at the edge of both the GinOE and GinUE, but essentially different in the depletion region of the GinOE.

Independent component model (ICM) is a widely-used population distribution in high-dimensional data analysis and random matrix theory. In this work, we study the kurtosis of the ICM, which is an important parameter in asymptotic distributions of many commonly used statistics and is also an important criterion for measuring the heavy-tailed nature of the data. Based on U-statistics, we develop an estimation method. Theoretically, we show that the proposed estimator is consistent under regular conditions, especially we relax the restriction that the data dimension and the sample size are of the same order. Furthermore, we derive the asymptotic normality of the estimator, which allows us to construct confidence intervals and hypothesis testing statistics. Computationally, we provide a fast and efficient algorithm by leveraging the matrix structure, where the computational complexity is essentially equivalent to computing the sample covariance matrix and the Gram matrix. Finally, the effectiveness of the proposed method is demonstrated through simulated data and real-world data.

We consider pairs of anti-commuting $2p$-by-$2p$ Hermitian matrices that are chosen randomly with respect to a Gaussian measure. Generically such a pair decomposes into the direct sum of $2$-by-$2$ blocks on which the first matrix has eigenvalues $\pm x_j$ and the second has eigenvalues $\pm y_j$. We call $\{(x_j,y_j)\}$ the skew spectrum of the pair. We derive a formula for the probability density of the skew spectrum, and show that the elements are repelling.

In this paper, we consider an initial $n\times d$ random matrix with non-Gaussian correlated entries on each row and independent entries from one row to another. The correlation on the rows is given by the correlation of the increments of the Rosenblatt process, which is a non-Gaussian self-similar process with stationary increments, living in the second Wiener chaos. To this initial matrix, we associate a Wishart tensor of length $p\ge 2$. We study the limit behavior in distribution of this Wishart tensor in the high-dimensional regime, i.e. when $n,d$ are large enough. We prove that the vector corresponding to the $p$-Wishart tensor converges to an $n^p$-dimensional non-Gaussian vector, with Rosenblatt random variables on its hyper-diagonals and zeros outside the hyper-diagonals.

It is well known that, under some assumptions, the limit distribution of block random matrices and their partial transposition converges to the distributions of random variables in some non-commutative probability spaces. Using free probability theory, we obtain the relation between the free cumulants of the corresponding random variables. As an application, we can derive a new family of co-completely positive and $k$-positive maps by using the Wishart ensemble.

Let $W_n$ be a Wigner matrix of dimension $n\times n$ with eigenvalues ${\lambda }_{1,n}\ge \cdots \ge {\lambda }_{n,n}$ and $g$ be an analytic function on $[-2-𝜀,2+𝜀]$ with polynomial growth. It is known that Tr[$g(W_n$)]-E[Tr[$g(W_n$)]] converges in distribution to a normal random variable with mean $0$ and a finite variance depending on $g$. On the other hand, it is also known that $n^{2/3}({\lambda }_{1,n}-2)$ converges in distribution to the GOE Tracy widom law. In this paper we prove that whenever the entries of the Wigner matrix are sub-Gaussian, Tr[$g(W_n$)]-E[Tr[$g(W_n$)]] is asymptotically independent of the point process at the edge of the spectrum. Hence, one gets that $n^{2/3}({\lambda }_{1,n}-2)$ and Tr[$g(W_n$)]-E[Tr[$g(W_n$)]] are asymptotically independent. The main ingredient of the proof is based on a recent paper by Banerjee [A new combinatorial approach for tracy–widom law of wigner matrices, preprint (2022), arXiv:2201.00300]. The result of this paper can be viewed as a first step to find the joint distribution of eigenvalues in the bulk and the edge.

In this paper, using the idea of grouping under moderate data framework, we propose the median-of-means-type nonparametric estimator for Pearson’s correlation coefficient which has been used widely in various disciplines. Under certain conditions on the growing rate of the number of subgroups, the consistency and asymptotic normality of the proposed estimator are investigated. Furthermore, we construct a new method to test Pearson’s correlation coefficient based on the empirical likelihood method for median. Extensively numerical simulations are designed to demonstrate the superiorities of our estimator. It is shown that the new proposed estimator is robust with respect to outliers. Finally, we use the proposed method to study Pearson’s correlation between open price and the rate of price spread for the Shanghai Stock Exchange composite index from 18 May 2015 to 21 June 2019.