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

Let X1,…,Xn be independent and identically distributed random vectors in Rd. Suppose EX1=0, Cov(X1)=Id, where Id is the d×d identity matrix. further that there exist positive constants t0 c0 such Eet0|X1|≤c0<∞, |⋅| denotes Euclidean norm. W= ∑i=1nXi∕n let Z a d-dimensional standard normal vector. Q symmetric definite matrix whose largest eigenvalue 1. We prove for 0≤x≤εn1∕6, P(|Q1∕2W|>x) P(|Q1∕2Z|>x)−1≤C1+x5 det(Q1∕2)n+x6 nford≥5 P(|Q1∕2Z|>x)−1≤C1+x3 det(Q1∕2)n d d+1+x6 nfor1≤d≤4, ε C are depending only on d,t0, c0. This first extension of Cramér-type moderate deviation to multivariate setting with faster convergence rate than 1∕n. The range x=o(n1∕6) relative error vanish dimension requirement d≥5 1∕n both optimal. our result using new change measure, two-term Edgeworth expansion changed cancellation by symmetry terms order