  We study the estimation of high-dimensional covariance matrices under
elliptical factor models with 2 + {\epsilon}th moment. For such heavy-tailed
data, robust estimators like the Huber-type estimator in Fan, Liu and Wang
(2018) can not achieve sub-Gaussian convergence rate. In this paper, we develop
an idiosyncratic-projected self-normalization (IPSN) method to remove the
effect of heavy-tailed scalar parameter, and propose a robust pilot estimator
for the scatter matrix that achieves the sub-Gaussian rate. We further develop
an estimator of the covariance matrix and show that it achieves a faster
convergence rate than the generic POET estimator in Fan, Liu and Wang (2018).
