Cramer-type moderate deviation for quadratic forms with a fast rate

Let X-1,..., X-n be independent and identically distributed random vectors in R-d. Suppose EX1 = 0, Cov(X-1) = (I)d, where I-d is the d x d identity matrix. Suppose further that there exist positive constants t(0) and c(0) such that Ee(t0 vertical bar X1|) <= c(0) < infinity, where vertical bar center dot vertical bar denotes the Euclidean norm. Let W = Sigma(n)(i=1) X-i root n and let Z be a d-dimensional standard normal random vector. Let Q be a d x d symmetric positive definite matrix whose largest eigenvalue is 1. We prove that for 0 <= x <= epsilon n(1/6), vertical bar P(|Q(1/2)W vertical bar > x)/P(vertical bar Q(1/2)Z vertical bar > x) - 1 vertical bar <= C (1 + x(5)/det (Q(1/2))n + x(6)/n) for d >= 5 and vertical bar P(|Q(1/2)W vertical bar > x)/P(vertical bar Q(1/2)Z vertical bar > x) - 1 vertical bar <= C (1 + x(3)/det (Q(1/2))n(d/d+1) + x(6)/n) for 1 <= d <= 4, where epsilon and C are positive constants depending only on d, t(0), and c(0). This is a first extension of Cramer-type moderate deviation to the multivariate setting with a faster convergence rate than 1/root n. The range of x = o(n(1/6)) for the relative error to vanish and the dimension requirement d >= 5 for the 1/n rate are both optimal. We prove our result using a new change of measure, a two-term Edgeworth expansion for the changed measure, and cancellation by symmetry for terms of the order 1/root n.