Asymptotic Distribution of Quadratic Forms and Applications

We consider the quadratic forms Q = (sigma) a j X j X k+ (sigma) ajj (X j 2 -E X j 2 )where X j are i.i.d. random variables with finite sixth moment. For a large class of matrices (a jk ) the distribution of Q can be approximated by the distribution of a second order polynomial in Gaussian random variables. We provide optimal bounds for the Kolmogorov distance between these distributions, extending previous results for matrices with zero diagonals to the general case. Furthermore, applications to quadratic forms of ARMA-processes, goodness-of-fit as well as spacing statistics are included.