  Correlation matrices play a key role in many multivariate methods (e.g.,
graphical model estimation and factor analysis). The current state-of-the-art
in estimating large correlation matrices focuses on the use of Pearson's sample
correlation matrix. Although Pearson's sample correlation matrix enjoys various
good properties under Gaussian models, it is not an effective estimator when
facing heavy-tailed distributions. As a robust alternative, Han and Liu [J. Am.
Stat. Assoc. 109 (2015) 275-287] advocated the use of a transformed version of
the Kendall's tau sample correlation matrix in estimating high dimensional
latent generalized correlation matrix under the transelliptical distribution
family (or elliptical copula). The transelliptical family assumes that after
unspecified marginal monotone transformations, the data follow an elliptical
distribution. In this paper, we study the theoretical properties of the
Kendall's tau sample correlation matrix and its transformed version proposed in
Han and Liu [J. Am. Stat. Assoc. 109 (2015) 275-287] for estimating the
population Kendall's tau correlation matrix and the latent Pearson's
correlation matrix under both spectral and restricted spectral norms. With
regard to the spectral norm, we highlight the role of "effective rank" in
quantifying the rate of convergence. With regard to the restricted spectral
norm, we for the first time present a "sign sub-Gaussian condition" which is
sufficient to guarantee that the rank-based correlation matrix estimator
attains the fast rate of convergence. In both cases, we do not need any moment
condition.
