Asymptotic expansions in non-central limit theorems for quadratic forms

Götze F, Tikhomirov AN. Asymptotic expansions in non-central limit theorems for quadratic forms. JOURNAL OF THEORETICAL PROBABILITY. 2005;18(4):757-811.We consider quadratic forms of the type [GRAPHICS] where X-j are i.i.d. random variables with common distribution F and finite fourth moment, A = {a(jk)}(j,k=1)(N) denotes a symmetric matrix with eigenvalues lambda(1),...,lambda(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 lambda(13) for the approximation of the distribution of Q(F, A) by the distribution of Q(Phi, A) where Phi is standard Gaussian distribution. In typical cases this error is of optimal order O(N-1)