t_1 I)\}>\epsilon \Big \},\end{aligned}$$

and $t_1$ is average of the diagonal elements of $\hat{R}_K^{-1}$. The minimization in (3.6) over $Z$ is for fixed $ (\lambda, \gamma) \in \hat{\mathscr{S}}^{{K}}_2$. The proposed JPEN estimator of inverse covariance matrix (based on regularized inverse correlation matrix estimator $\hat{Z}_{K}$) is given by $\hat{\Omega}_{{K}}=(S^+)^{-1/2}{\hat{Z}}_{{K}}(S^+)^{-1/2}$, where $S^+$ is a diagonal matrix of the diagonal elements of $S$. Moreover (3.6) is a convex optimization problem and $\hat{Z}_K$ is positive definite.\
Next we state the consistency of estimators $\hat{Z}_{{K}}$ and $\hat{\Omega}_{{K}}$.

Under Assumptions B0, B1, B2 and for $(\lambda,\gamma) \in \hat{\mathscr{S}}^{{K}}_{2}$, $$\|\hat{Z}_{{K}}-R_0^{-1}\|_F=O_P \Big ( \sqrt{\frac{s~log~p}{n}} \Big )  ~~~\text{and} ~~~\|\hat{\Omega}_{{K}}-\Omega_0\|=O_P \Big ( \sqrt{\frac{(s+1)~log~p}{n}} \Big )$$ where $R_0^{-1}$ is the inverse of true correlation matrix.

**Remark:1.** Note that the JPEN estimator $\hat{\Omega}_{{K}}$ achieves minimax rate of convergence for the class of covariance matrices satisfying assumption $B0$, $B1$, and $B2$ and therefore optimal. The similar rate is obtained in @Cai2:40 for their class of sparse inverse covariance matrices.\
Next we give another estimate of inverse covariance matrix based on $\hat{\Sigma}_{S}$. Consider the following optimization problem: $$\hat{\Omega}_S=\operatorname*{arg\,min}_{\Omega=\Omega^T,tr(\Omega)=tr(\hat{\Sigma}_{S}^{-1})|(\lambda, \gamma) \in \hat{\mathscr{S}}^{{S}}_2}~~\Big[  ||\Omega- \hat{\Sigma}_{S}^{-1}||^2_F +  \lambda \|\Omega^-\|_1  + \gamma\sum_{i=1}^{p} \{\sigma_i(\Omega)-\bar{\sigma}_{\Omega}\}^2\Big],$$ where

$$\begin{aligned}
\hat{\mathscr{S}}^{{S}}_2&= \Big \{(\lambda,\gamma): \lambda,\gamma >0, \lambda \asymp \gamma \asymp \sqrt{\frac{log~ p}{n}}, ~\forall \epsilon 