)$, with $ \hat{\Sigma}_{{\widehat{S}}}$ given in \[eq:Sigma.loco\]. Notice that $\hat{\Sigma}_{{\widehat{S}}}$ is almost surely positive definite, a consequence of adding extra noise in the definition of $\gamma_{{\widehat{S}}}$ and $\hat{\gamma}_{{\widehat{S}}}$. Then, using Theorem 2.1 in [@cherno2], there exists a universal constant $C > 0$ such that $$\label{eq::secondx}
    \sup_{ t = (t_j, j \in {\widehat{S}}) \in \mathbb{R}^{{\widehat{S}}}_{+}} \Bigl| \mathbb{P}(
    \sqrt{n}|\hat\gamma_{{\widehat{S}}}(j)  - \gamma_{{\widehat{S}}}(j) | \leq t_j, \forall j \in {\widehat{S}}) - 
    \mathbb{P}(|Z_n(j)|  \leq t_j, \forall j \in {\widehat{S}})\Bigr| \leq C
    \mathrm{E}_{1,n},$$ where $\mathrm{E}_{1,n}$ is given in . By restricting the supremum in the above display to all $t \in \mathbb{R}^{{\widehat{S}}}_+$ with identical coordinates, we also obtain that $$\label{eq::firstx}
    \sup_{t > 0} \Bigl| \mathbb{P}(\sqrt{n}||\hat\gamma_{{\widehat{S}}} - \gamma_{{\widehat{S}}}||_\infty \leq t) - 
    \mathbb{P}\left(||Z_n||_\infty \leq t \right)\Bigr| \leq C \mathrm{E}_{1,n}.$$ In order to show and , we will use the same arguments used in the proofs of and . We first define $\mathcal{E}_n$ to be the event that $$\label{eq:loco.aleph}
\max_{i,j} \left| \widehat{\Sigma}_{{\widehat{S}}}(i,j) - \Sigma_{{\widehat{S}}}(i,j) \right| \leq
N_n,$$ where $N_n$ is as in . Each entry of $\widehat{\Sigma}_{{\widehat{S}}} - \Sigma_{{\widehat{S}}}$ is bounded in absolute value by $\left( 2(A+\tau) + \epsilon \right)^2$, and therefore is a sub-Gaussian with parameter $\left( 2(A+\tau) + \epsilon \right)^4$. Using a standard derivation for bounding the maximum of sub-Gaussian random variables we obtain that $\mathbb{P}(\mathcal{E}_n^c) \leq \frac{1}{n} $. The bound follows from the same arguments as in the proof : combine the Gaussian comparison Theorem \[thm:comparisons\] with and notice that $\epsilon/\sqrt{3}$ is a lower bound on the standard deviation of the individual coordinates of the $\delta_i$’s. In particular, the Gauss