norm is $c_{n,p} \sqrt{log~p/n}$ which is same as that of $\hat{\Sigma}_K$ for $1 \leq c_{n,p}=\sqrt{s}$.\
[**2.**]{} @Bickel1:4 proved that under the assumption of $\sum_{j=1}|\sigma_{ij}|^q \leq c_0(p)$ for some $ 0 \leq q \leq 1$, the hard thresholding estimator of the sample covariance matrix for tuning parameter $\lambda \asymp \sqrt{(log~p)/n}$ is consistent in operator norm at a rate no worse than $ O_P\Big ( c_0(p) \sqrt{p}(\frac{log ~p}{n})^{(1-q)/2} \Big ) $ where $c_0(p)$ is the upper bound on the number of non-zero elements in each row. Here the truly sparse case corresponds to $q=0$. The rate of convergence of $\hat{\Sigma}_K$ is same as that of @Bickel1:4 except in the following cases:\
[**Case (i)**]{} The covariance matrix has all off diagonal elements zero except last row which has $ \sqrt{p}$ non-zero elements. Then $c_0(p)=\sqrt{p}$ and $ \sqrt{s}=\sqrt{2~\sqrt{p}-1}$. The opeartor norm rate of convergence for JPEN estimator is $O_P \Big ( \sqrt{\sqrt{p}~(log~p)/n} \Big )$ where as rate of Bickel and Levina’s estimator is $O_P \Big (\sqrt{p~(log~p)/n} \Big )$.\
[**Case (ii)**]{} When the true covariance matrix is tridiagonal, we have $c_0(p)=2$ and $s=2p-2$, the JPEN estimator has rate of $\sqrt{p~log~p/n}$ whereas the Bickel and Levina’s estimator has rate of $\sqrt{log~p/n}$.\
For the case $\sqrt{s} \asymp 