and positive). A well-conditioned means that \[@Ledoit:17)\] inverting the matrix does not explode the estimation error. Assumption A1 is more of a definition which says that the number of non-zero off diagonal elements are bounded by some positive integer. Theorem 3.1 gives the rate of convergence of the proposed correlation based covariance matrix estimator (2.4). The following Lemmas show that optimization problem in (2.4) is convex and the proposed JPEN estimator (2.4) is positive definite.

The optimization problem in (2.4) is convex.

The estimator given by (2.4) is positive definite for any $2 \le n < \infty $ and $p <\infty$.

Let $(\lambda, \gamma) \in \hat{\mathscr{S}}^{K}_1$ and $\hat{\Sigma}_K$ be as defined in (2.4). Under Assumptions A0, A1, A2, $$\|\hat{R}_K-R_0\|_F=O_P \Big ( \sqrt{\frac{s~ log~p}{n}} \Big ) ~~~ \text{and} ~~~\|\hat{\Sigma}_K-\Sigma_0\|=O_P \Big ( \sqrt{\frac{(s+1) log~p}{n}} \Big ),$$ where $R_0$ is true correlation matrix.

**Remark: 1.** The JPEN estimator $\hat{\Sigma}_K$ is minimiax optimal under the operator norm. In (@Cai2:40), the authors obtain the minimax rate of convergence in the operator norm of their covariance matrix estimator for the particular construction of parameter space $\mathscr{H}_0(c_{n,p}):=\Big \{ \Sigma : max_{1 \le i \le p}\sum_{i=1}^{p}I\{\sigma_{ij}\neq 0\} \leq c_{n,p} \Big \}$. They show that this rate in operator 