Random Matrices: Theory and Applications

The paper “Note sur une relation entre les intégrales définies des produits des fonctions” by C. Andréief is an often cited paper in random matrix theory, due to it containing what is now referred to as Andréief’s integration formula. Nearly all citing works state the publication year as 1883. However, the journal containing the paper, Mémories de la Societé des Sciences physiques et naturelles de Bordeaux, issue 3 volume 2 actually appeared in 1886. In addition to clarifying this point, some historical information relating to C. Andréief (better known as K. A. Andreev) and the lead up to this work is given, as is a review of some of the context of Andréief’s integration formula.

Recently Burkhardt et al. introduced the $k$-checkerboard random matrix ensembles, which have a split limiting behavior of the eigenvalues (in the limit all but $k$ of the eigenvalues are on the order of $\sqrt{N}$ and converge to semi-circular behavior, with the remaining $k$ of size $N$ and converging to hollow Gaussian ensembles). We generalize their work to consider non-Hermitian ensembles with complex eigenvalues; instead of a blip new behavior is seen, ranging from multiple satellites to annular rings. These results are based on moment method techniques adapted to the complex plane as well as analysis of singular values.

We introduce a new method to qualify the goodness of fit parameter estimation of compound Wishart models. Our method is based on the free deterministic equivalent $Z$-score, which we introduce in this paper. Furthermore, an application to two-dimensional autoregressive moving-average model is provided.

Our proposed method is a generalization of statistical hypothesis testing to one-dimensional moving average model based on fluctuations of real compound Wishart matrices by Hasegawa et al. [A. Hasegawa, N. Sakuma and H. Yoshida, Fluctuations of Marchenko–Pastur limit of random matrices with dependent entries, Statist. Probab. Lett. 127 (2017) 85–96].

We show that independent elliptic matrices converge to freely independent elliptic elements. Moreover, the elliptic matrices are asymptotically free with deterministic matrices under appropriate conditions. We compute the Brown measure of the product of elliptic elements. It turns out that this Brown measure is same as the limiting spectral distribution.