Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms

Abstract

We consider quadratic forms of the type $$ Q(F,{\bf A})=\sum_{\mathop{1\le j,k \le N}\limits_{j\ne k}}a_{jk} X_j X_k, $$ where X j are i.i.d. random variables with common distribution F and finite fourth moment, $${\bf A}=\{a_{jk}\}_{j,k=1}^N$$ denotes a symmetric matrix with eigenvalues λ1, ..., λ N ordered to be non-increasing in absolute value. We prove explicit bounds in terms of sums of 4th powers of entries of the matrix A and the size of the eigenvalue λ13 for the approximation of the distribution of Q(F,A) by the distribution of Q (φ, A) where φ is standard Gaussian distribution. In typical cases this error is of optimal order $${\cal {O}}(N^{-1})$$