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

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.

De-Biased Graphical Lasso for High-Frequency Data

“…In the case that d is fixed, we typically resolve this issue by proving stable convergence in law of ; see e.g., [ 26 ] for details. However, extension of this approach to the case that as is far from trivial as discussed at the beginning of [ 20 ] (Section 3). For this reason, [ 20 ] gives a result to directly establish Equation ( 5 ) type convergence in a high-dimensional setting.…”

“…However, extension of this approach to the case that as is far from trivial as discussed at the beginning of [ 20 ] (Section 3). For this reason, [ 20 ] gives a result to directly establish Equation ( 5 ) type convergence in a high-dimensional setting. This result will be used in Section 4 to apply our abstract theory to a more concrete setting.…”

“…Despite the recent advances in this topic as above, most studies in this area focus only on point estimation of covariance and precision matrices, and there are little work about interval estimation and hypothesis testing for these objects. A few exceptions are [ 18 , 19 , 20 ]. The first two articles are concerned with continuous-time factor models: Kong and Liu [ 18 ] propose a test for the constancy of the factor loading matrix, while Pelger [ 19 ] assumes constant loadings and develops an asymptotic distribution theory to make inference for the factors and loadings.…”

“…The first two articles are concerned with continuous-time factor models: Kong and Liu [ 18 ] propose a test for the constancy of the factor loading matrix, while Pelger [ 19 ] assumes constant loadings and develops an asymptotic distribution theory to make inference for the factors and loadings. Meanwhile, Koike [ 20 ] establishes a high-dimensional central limit theorem for the realized covariance matrix which allows us to construct simultaneous confidence regions or carry out multiple testing for entries of the high-dimensional covariance matrix of high-frequency data. The aim of this study is to develop such a statistical inference theory for the precision matrix of high-frequency data.…”

“…case. In a low-dimensional setting, this issue is typically resolved by the concept of stable convergence (see e.g., [ 26 ]), but the applicability of this approach is questionable in our setting due to the high-dimensionality (see pages 1451–1452 of [ 20 ] for a discussion). Instead, we rely on the recent high-dimensional central limit theory of [ 20 ] to establish the asymptotic distribution theory for the de-biased estimator, where we settle the above difficulty with the help of Malliavin calculus.…”