In Section \[sec:rc\] we apply the developed theory to establish the asymptotic mixed normality of realized covariance matrices in a high-dimensional setting and illustrate its application to testing the residual sparsity of a continuous-time factor model. Section \[sec:simulation\] provides a small simulation study as well as an empirical illustration using real data. All the proofs are collected in the Appendix.

Preliminaries {#sec:preliminaries}
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In this section we present some notation and concepts used throughout the paper.

Basic notation {#sec:basic}
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We begin by introducing some basic notation which more or less common in the literature. For a vector $x\in\mathbb{R}^d$, we write the $i$-th component of $x$ as $x^i$ for $i=1,\dots,d$. Also, we set $\min x:=\min_{1\leq i\leq d}x^i$. For two vectors $x,y\in\mathbb{R}^d$, the statement $x\leq y$ means $x^i\leq y^i$ for all $i=1,\dots,d$. For a vector $x\in\mathbb{R}^d$ and a scalar $a\in\mathbb{R}$, we set $$x\pm a:=(x^1\pm a,\dots,x^d\pm a)^\top.$$ Here, $\top$ stands for the transpose of a matrix.

For a matrix $A$, we write its $(i,j)$-th entry as $A^{ij}$. Also, $A^{i\cdot}$ and $A^{\cdot j}$ denote the $i$-th row vector and the $j$-th column vector, respectively. Here, we regard both the vectors $A^{i\cdot}$ and $A^{\cdot j}$ as column vectors. If $A$ is an $m\times d$ 