On spectral distribution of high dimensional covariation matrices

TL;DR

This paper develops an asymptotic theory for the spectral distribution of high-dimensional covariation matrices derived from Brownian diffusions, considering high-frequency data and the limit where dimension and sample size grow proportionally.

Contribution

It introduces a new asymptotic framework for spectral distributions of covariation matrices with time-varying integrands in high dimensions, using the method of moments and graph theory.

Findings

Spectral distribution converges almost surely under certain conditions.

Provides a theoretical basis for high-dimensional covariance analysis.

Extends spectral analysis to time-varying integrands in Brownian diffusions.

Abstract

In this paper we present the asymptotic theory for spectral distributions of high dimensional covariation matrices of Brownian diffusions. More specifically, we consider $N$-dimensional Ito integrals with time varying matrix-valued integrands. We observe $n$ equidistant high frequency data points of the underlying Brownian diffusion and we assume that $N/n\rightarrow c\in (0,\infty)$. We show that under a certain mixed spectral moment condition the spectral distribution of the empirical covariation matrix converges in distribution almost surely. Our proof relies on method of moments and applications of graph theory.