We establish the asymptotic normality of a quadratic form Qn in martingale difference random variables ηt 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 ηt, asymptotic normality holds under condition | | A| | sp= o(| | A| |) , where | | A| | sp and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences ηt has been an important open problem. We provide such sufficient conditions in this paper.
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