---
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.

    *Keywords*: Bootstrap; Chernozhukov-Chetverikov-Kato theory; High-dimensions; High-frequency data; Malliavin calculus; Multiple testing.
author:
- 'Yuta Koike[^1] [^2] [^3] [^4]'
bibliography:
- 'base.bib'
title: 'Mixed-normal limit theorems for multiple Skorohod integrals in high-dimensions, with application to realized covariance'
---

Introduction
============

Covariance matrix estimation of multiple assets is one of the most active research areas in high-frequency financial econometrics. Recently, many authors have been attacking the high-dimensionality in covariance matrix estimation from high-frequency data. A pioneering work on this topic is the paper by @WZ2010, where the regularization methods (banding and thresholding) proposed in @BL2008band [@BL2008th] have been applied to estimating high-dimensional quadratic covariation matrices from noisy and non-synchronous high-frequency data. Subsequently, their approach has been 