  Moment inequality for quadratic forms of random vectors is of particular
interest in covariance matrix testing and estimation problems. In this paper,
we prove a Rosenthal-type inequality, which exhibits new features and certain
improvement beyond the unstructured Rosenthal inequality of quadratic forms
when dimension of the vectors increases without bound. Applications to test the
block diagonal structures and detect the sparsity in the high-dimensional
covariance matrix are presented.
