and even if it is regular, computation of the inverse is typically time-consuming. Nevertheless, we can fortunately show that, in order to estimate quantiles of the unconditional distribution $\Xi_n\mathfrak{Z}_n$, it is sufficient to only estimate its $\mathcal{F}$-conditional distribution. We remark that this fact has already been known in high-frequency financial econometrics and typically been used to construct jump-related testing procedures; see [@JT2009; @LTT2017] for example. Formally, we can prove the following result:

\[prop:quantile\] For each $n\in\mathbb{N}$, let $T_n,T_n^\dagger,T_n^*$ be random variables defined on an extension of the probability space $(\Omega,\mathcal{F},P)$. Suppose that $$\begin{aligned}
&\sup_{x\in\mathbb{R}}\left|P(T_n\leq x)-P(T_n^\dagger\leq x)\right|\to0,
&\sup_{x\in\mathbb{R}}\left|P(T_n^*\leq x|\mathcal{F})-P(T_n^\dagger\leq x|\mathcal{F})\right|\to^p0\end{aligned}$$ as $n\to\infty$. Suppose also that there is a sequence $(E_n)$ of elements in $\mathcal{F}$ such that the $\mathcal{F}$-conditional distribution of $T_n^\dagger$ has the density on $E_n$ for every $n$ and $\lim_{n\to\infty}P(E_n)=1$. For each $n\in\mathbb{N}$, let $q_n^*$ be the $\mathcal{F}$-conditional quantile function of $T_n^*$: $$q_n^*(\alpha)=\inf\{x\in\mathbb{R}:P(T_n^*\leq x|\mathcal{F})\geq \alpha\},\qquad \alpha\in(0,1).$$ Then we have $$P(T_n \leq q_n^*(\alpha))\to\alpha$$ as $n\to\infty$ for all $\alpha\in(0,1)$.

Application to realized covariance {#sec:rc}
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In this section we assume that the probability space $(\Omega,\mathcal{F},P)$ admits the structure such that $\Omega=\Omega'\times\mathbf{W}$, $\mathcal{F}=\mathcal{F}'\otimes\mathbf{B}$ and $P=P'\times\mathbf{P}$ for some probability space $(\Omega',\mathcal{F}',P')$ and the $r$-dimensional Wiener space $(\mathbf{W},\mathbf{B},\mathbf{P})$ over time interval $[0,1]$, and consider the 