Spectra of large time-lagged correlation matrices from Random Matrix Theory

TL;DR

This paper uses advanced random matrix theory techniques to analyze the spectral properties of large time-lagged correlation matrices, providing new insights into their eigenvalue distributions and eigenvector correlations.

Contribution

It introduces a general solution for the spectrum of matrices of the form (1/T)XAX† with arbitrary A, and applies it to large lagged correlation matrices, extending existing spectral analysis methods.

Findings

Spectral features vary with time-lag depth.

Eigenvector correlations exhibit specific properties.

Results are verified through numerical simulations.

Abstract

We analyze the spectral properties of large, time-lagged correlation matrices using the tools of random matrix theory. We compare predictions of the one-dimensional spectra, based on approaches already proposed in the literature. Employing the methods of free random variables and diagrammatic techniques, we solve a general random matrix problem, namely the spectrum of a matrix $\frac{1}{T}XAX^{\dagger}$, where $X$ is an $N\times T$ Gaussian random matrix and $A$ is \textit{any} $T\times T$, not necessarily symmetric (Hermitian) matrix. As a particular application, we present the spectral features of the large lagged correlation matrices as a function of the depth of the time-lag. We also analyze the properties of left and right eigenvector correlations for the time-lagged matrices. We positively verify our results by the numerical simulations.