  We consider asymptotic distributions of maximum deviations of sample
covariance matrices, a fundamental problem in high-dimensional inference of
covariances. Under mild dependence conditions on the entries of the data
matrices, we establish the Gumbel convergence of the maximum deviations. Our
result substantially generalizes earlier ones where the entries are assumed to
be independent and identically distributed, and it provides a theoretical
foundation for high-dimensional simultaneous inference of covariances.
