Asymptotic normality of quadratic forms of martingale differences

We establish the asymptotic normality of a quadratic form (Q_n) in martingale difference random variables (eta _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), 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 (eta _t) has been an important open problem. We provide such sufficient conditions in this paper.