A note on moment inequality for quadratic forms

Abstract

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.

Variance inequalities for quadratic forms with applications

“…when each X t has zero mean. By Lemma 1 of Andrews [1], (5) holds when (X t ) ∞ t=−∞ is a strongly mixing sequence with mixing coefficients…”

“…When random variables {X i } n i=1 are independent, moment inequalities for quadratic forms n i,j=1 a ij X i X j are well-studied (e.g., see Lemma B.26 in Bai and Silverstein [4] and Chen [5]). In the time series context, similar inequalities were obtained by many authors in connection with spectral density estimation and long-run variance estimation (e.g., see…”