  We consider the estimation of integrated covariance (ICV) matrices of high
dimensional diffusion processes based on high frequency observations. We start
by studying the most commonly used estimator, the realized covariance (RCV)
matrix. We show that in the high dimensional case when the dimension $p$ and
the observation frequency $n$ grow in the same rate, the limiting spectral
distribution (LSD) of RCV depends on the covolatility process not only through
the targeting ICV, but also on how the covolatility process varies in time. We
establish a Mar\v{c}enko--Pastur type theorem for weighted sample covariance
matrices, based on which we obtain a Mar\v{c}enko--Pastur type theorem for RCV
for a class $\mathcal{C}$ of diffusion processes. The results explicitly
demonstrate how the time variability of the covolatility process affects the
LSD of RCV. We further propose an alternative estimator, the time-variation
adjusted realized covariance (TVARCV) matrix. We show that for processes in
class $\mathcal {C}$, the TVARCV possesses the desirable property that its LSD
depends solely on that of the targeting ICV through the Mar\v{c}enko--Pastur
equation, and hence, in particular, the TVARCV can be used to recover the
empirical spectral distribution of the ICV by using existing algorithms.
