Mixed-normal limit theorems for multiple Skorohod integrals in high-dimensions, with application to realized covariance

TL;DR

This paper develops mixed-normal limit theorems for high-dimensional multiple Skorohod integrals and applies them to analyze the asymptotic behavior of realized covariance matrices in high-frequency financial data, enabling new high-dimensional inference methods.

Contribution

It introduces a novel mixed-normal approximation framework for high-dimensional Skorohod integrals and applies it to establish the asymptotic distribution of realized covariance matrices.

Findings

Asymptotic mixed normality of realized covariance in high dimensions.

Effective testing procedure for residual sparsity in factor models.

Theoretical results hold even when dimension exceeds sample size.

Abstract

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.