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In this paper we present the asymptotic theory for spectral distributions ofhigh dimensional covariation matrices of Brownian diffusions. Morespecifically, we consider $N$-dimensional Ito integrals with time varyingmatrix-valued integrands. We observe $n$ equidistant high frequency data pointsof 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 thespectral distribution of the empirical covariation matrix converges indistribution almost surely. Our proof relies on method of moments andapplications of graph theory.

On spectral distribution of high dimensional covariation matrices