Random Matrices: Theory and Applications

We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix $\tilde{W}$ and its minor $W$. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of $\tilde{W}$ and $W$. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices, the fluctuation may entirely vanish.

In this paper, we decompose the volatility of a diffusion process into systematic and idiosyncratic components, which are not identified with observations discretely sampled from univariate process. Using large dimensional high-frequency data and assuming a factor structure, we obtain consistent estimates of the Laplace transforms of the systematic and idiosyncratic volatility processes. Based on the discrepancy between realized bivariate Laplace transform of the pair of systematic and idiosyncratic volatility processes and the product of the two marginal Laplace transforms, we propose a Kolmogorov–Smirnov-type independence test statistics for the two components of the volatility process. A functional central limit theorem for the discrepancy is established under the null hypothesis that the systematic and idiosyncratic volatilities are independent. The limiting Gaussian process is realized by a simulated discrete skeleton process which can be applied to define an approximate critical region for an independence test.

This paper develops an outlier detection procedure for multinomial data when the number of categories tends to infinity. Most of the outlier detection methods are based on the assumption that the observations follow multivariate normal distribution, while in many modern applications, the observations either are measured on a discrete scale or naturally have some categorical structures. For such multinomial observations, there are rather limited approaches for outlier detection. To overcome the main obstacle, the least trimmed distances estimator for multinomial data and a fast algorithm to identify the clean subset are introduced in this work. Also, a threshold rule is considered through the asymptotic distribution of measure distance to identify outliers. Furthermore, a one-step reweighting scheme is proposed to improve the efficiency of the procedure. Finally, the finite sample performance of our method is evaluated through simulations and is compared with that of available outlier detection methods.

In this paper, we propose some new tests for high-dimensional covariance matrices that are applicable to generally distributed populations with finite fourth moments. The proposed test statistics are the maximum of the likelihood ratio test statistic and the statistic based on the Frobenius norm. The advantage of the new tests is the good performance in terms of power for both the traditional case, in which the dimension is much smaller than the sample size, and the high-dimensional case, in which the dimension is large compared to the sample size. In the one-sample case, the new test is proposed for testing the hypothesis that the high-dimensional covariance matrix equals an identity matrix. In the two-sample case, the new test is developed for testing the equality of two high-dimensional covariance matrices. By using the random matrix theory, the asymptotic distributions of the proposed new tests are derived under the assumption that the dimension and the sample size proportionally tend toward infinity. Finally, numerical studies are conducted to investigate the finite sample performance of the proposed new tests.

We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes of combinatorial maps, for which we develop the combinatorial machinery necessary for the asymptotic study. Finally, we present some applications to the theory of random quantum states in quantum information theory.

We consider large-dimensional Hermitian or symmetric random matrices of the form $W=M+𝜗V$, where $M$ is a Wigner matrix and $V$ is a real diagonal matrix whose entries are independent of $M$. For a large class of diagonal matrices $V$, we prove that the fluctuations of linear spectral statistics of $W$ for $C^2$ test function can be decomposed into that of $M$ and of $V$, and that each of those weakly converges to a Gaussian distribution. We also calculate the formulae for the means and variances of the limiting distributions.