Asymptotic normality of quadratic forms of martingale differences

Liudas Giraitis (), Masanobu Taniguchi () and Murad S. Taqqu ()
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Liudas Giraitis: Queen Mary University of London
Masanobu Taniguchi: Waseda University
Murad S. Taqqu: Boston University

Statistical Inference for Stochastic Processes, 2017, vol. 20, issue 3, No 4, 315-327

Abstract: Abstract We establish the asymptotic normality of a quadratic form $$Q_n$$ Q n in martingale difference random variables $$\eta _t$$ η t when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables $$\eta _t$$ η t , asymptotic normality holds under condition $$||A||_{sp}=o(||A||) $$ | | A | | s p = o ( | | A | | ) , where $$||A||_{sp}$$ | | A | | s p and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences $$\eta _t$$ η t has been an important open problem. We provide such sufficient conditions in this paper.

Keywords: Asymptotic normality; Quadratic form; Martingale differences; 62E20; 60F05 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s11203-016-9143-3

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