  This paper develops mixed-normal approximations for probabilities that
vectors of multiple Skorohod integrals belong to random convex polytopes when
the dimensions of the vectors possibly diverge to infinity. We apply the
developed theory to establish the asymptotic mixed normality of the realized
covariance matrix of a high-dimensional continuous semimartingale observed at a
high-frequency, where the dimension can be much larger than the sample size. We
also present an application of this result to testing the residual sparsity of
a high-dimensional continuous-time factor model.
