Sparse PCA-based on high-dimensional Itô processes with measurement errors

This paper investigates the eigenspace estimation problem for the large integrated volatility matrix based on non-synchronized and noisy observations from a high-dimensional Itô process. We establish a minimax lower bound for the eigenspace estimation problem and propose sparse principal subspace estimation methods by using the multi-scale realized volatility matrix estimator or the pre-averaging realized volatility matrix estimator. We derive convergence rates of the proposed eigenspace estimators and show that the estimators can achieve the minimax lower bound, and thus are rate-optimal. The minimax lower bound can be established by Fano's lemma with an appropriately constructed subclass that has independent but not identically distributed normal random variables with zero mean and heterogeneous variances.