data without increasing $k$. **(ii)** Discriminant analysis \[@Mardia:19\]:, where the goal is to classify observations into different classes. Here estimates of covariance and inverse covariance matrices play an important role as the classifier is often a function of these entities. **(iii)** Regression analysis: If interest focuses on estimation of regression coefficients with correlated (or longitudinal) data, a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under mis-specification of the covariance structure. **(iv)** Gaussian graphical modeling \[@Mein:20, @Wainwright:27, @Yuan:34,@Yuan1:35\]:, the relationship structure among nodes can be inferred from inverse covariance matrix. A zero entry in the inverse covariance matrix implies conditional independence between the corresponding nodes.\
The estimation of large dimensional covariance matrix based on few sample observations is a difficult problem, especially when $n \asymp p$ (here $a_n \asymp b_n$ means that there exist positive constants $c$ and $C$ such that $c \le a_n/b_n \le C $). In these situations, the sample covariance matrix becomes unstable which explodes the estimation error. It is well known that the eigenvalues of sample covariance matrix are over-dispersed which means that the eigen-spectrum of sample 