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A note on moment inequality for quadratic forms

Affiliation:	1. Loughborough University, Department of Mathematical Sciences, Loughborough, Leicestershire, LE11 3TU, UK;2. University of Strathclyde, Department of Mathematics and Statistics, Glasgow, G1 1XH, UK;3. School of Economics and Management, Fuzhou University, China;1. Department of Statistics, Sungkyunkwan University, 25-2, Sungkyunkwan-ro, Jongno-gu, Seoul, 110-745, Republic of Korea;2. Mathematics Department, Tulane University, 6823 St. Charles Avenue, New Orleans, LA 70118, USA;3. Department of Statistics and Operations Research, UNC at Chapel Hill, CB#3260, Hanes Hall, Chapel Hill, NC 27599, USA;1. School of Sciences, Ningbo University of Technology, 201 Fenghua Rd., Ningbo 315211, PR China;2. Department of Mathematics, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, PR China

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.

Keywords:	Quadratic forms Rosenthal’s inequality High-dimensional covariance matrix
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