to apply a regularization procedure to estimating the quadratic covariation matrix $[Y,Y]_1=([Y^i,Y^j]_1)_{1\leq i,j\leq d}$ of $Y$, it is important to understand whether the target matrix is really sparse or not, and if so, how sparse it is. This amounts to evaluating the following series of the statistical hypotheses *simultaneously*: $$\label{test:sparse}
H_{(i,j)}: [Y^i,Y^j]_1\equiv0,\qquad i,j=1,\dots,d\text{ such that }i<j.$$ A natural way to construct test statistics for this problem is to estimate $[Y,Y]_1$ and test whether each of the entries is significantly away from 0 or not. Now suppose that $Y$ is observed at the equidistant times $t_h=h/n$, $h=0,1,\dots,n$. Then the most canonical estimator for $[Y,Y]_1$ would be the so-called *realized covariance matrix*: $$\label{def:rc}
\widehat{[Y,Y]}^n_1:=\sum_{h=1}^n(Y_{t_h}-Y_{t_{h-1}})(Y_{t_h}-Y_{t_{h-1}})^\top.$$ If one wants to test the null hypothesis such that all the hypotheses in is true, it is natural to consider the maximum type statistic $$\max_{(i,j)\in\Lambda}\left|\widehat{[Y,Y]}^n_1\right|,$$ where $\Lambda:=\{(i,j)\in\{1,\dots,d\}^2:i<j\}$. More generally, if one wants to control the family-wise error rate in multiple testing for the hypotheses , it is enough to approximate the distribution of $\max_{(i,j)\in\mathcal{L}}|\widehat{[Y,Y]}^n_1|$ for any $\mathcal{L}\subset\Lambda$, with the help of the stepdown procedure illustrated in @RW2005. Hence the problem amounts to approximating the distributions of such maximum type statistics in an appropriate sense. Using the test 