On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries

Abstract

For symmetric random matrices with correlated entries, which are functions of independent random variables, we show that the asymptotic behavior of the empirical eigenvalue distribution can be obtained by analyzing a Gaussian matrix with the same covariance structure. This class contains both cases of short and long range dependent random fields. The technique is based on a blend of blocking procedure and Lindeberg's method. This method leads to a variety of interesting asymptotic results for matrices with dependent entries, including applications to linear processes as well as nonlinear Volterra-type processes entries.

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

“…This convergence has been established in the Gaussian setup in [3,28,35]. In [9] it was extended by a comparison argument to the general setting we presented here.…”

“…As the dimension of H grows, its empirical spectral measure approaches [3,9,15,28,35] a non-random measure with density ρ defined through (1.3). In this setup m i solves (1.1) with s i j given by the Fourier transform of the correlation matrix (see Section 3.3).…”

On Structure Testing for Component Covariance Matrices of a High Dimensional Mixture

“…Therefore, it does not satisfy the Bai-Zhou weak dependence condition (7). Note that another weak dependence condition guaranteeing a limiting MP law is also available as in Banna et al (2015), but again this does not apply to the scale mixture (2).…”

Approximation to stable law by the Lindeberg principle

“…[8,7] obtained a general scaling of disordered system by expanding polynomial chaos. Besides a lot of applications in random matrices [4,6,26,41,29,42,1,5], Lindeberg principle has also been applied to other research 1 areas such as high dimensional regressions [14,15,20], time series [21,41,32,31], bootstrap [35,15], statistical learning [27,44] and so on.…”