Asymptotic expansion in the central limit theorem for quadratic forms

We consider the statistic of the form
$$Q_n=\sum_{j=1}^N a_{jj}(X_j^2-\mu_2)+\sum_{1\le j\ne k\le N}a_{jk}X_jX_k,$$
where $X_j$ are i.i.d. random variables with the finite sixth moment. We obtain the rate of convergence in the central limit theorem for one term Edgeworth expansion. Furthermore, applications to Toeplitz matrices, quadratic form of ARMA-processes, goodness-of-fit as well as spacing statistics are included.