worth mentioning that investigation of the spectral structure is also an issue in the context of high-dimensional covariance estimation. In high-frequency financial econometrics this was pioneered by @ZL2011, which investigates the problem of estimating the spectral distribution of the quadratic covariation estimation of a high-dimensional continuous Itô semimartingale from its high-frequency observation data. A similar problem has been studied in [@HP2014] under a different assumption on the model. Another important topic related to this issue is principal component analysis (PCA). In a high-dimensional setting, @KW2016 have studied PCA for integrated covariance matrices based on noisy and non-synchronous high-frequency data. Meanwhile, the recent work of @CMZ2018 investigates alternative PCA for high-frequency data based on the concept of realized eigenvalues, which was introduced in @AX2017jasa, in a high-dimensional setting.

To the best of the author’s knowledge, however, there is no work to establish a statistical inference theory validating simultaneous hypothesis testing and construction of uniformly valid confidence regions for high-dimensional quadratic covariation estimation from high-frequency data. Such a theory is important in statistical applications as illustrated by the following example: Let $Y=(Y_t)_{t\in[0,1]}$ be a $d$-dimensional continuous semimartingale. We denote by $Y^i$ the $i$-th component of $Y$ for every $i=1,\dots,d$. If one attempts 