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

Abstract

Let $X_1,\dots ,X_n$ be independent and identically distributed random vectors in ${\mathbb{R}}^d$. Suppose $\mathbb{E}{X}_1=0$, $\mathrm{Cov}(X_1)=I_d$, where $I_d$ is the $d\times d$ identity matrix. Suppose further that there exist positive constants $t_0$ and $c_0$ such that $\mathbb{E}{e}^{t_0|X_1|}\le c_0<\mathrm{\infty }$, where $|\cdot |$ denotes the Euclidean norm. Let $W={\sum }_{i=1}^n{X}_i∕\sqrt{n}$ and let Z be a d-dimensional standard normal random vector. Let Q be a $d\times d$ symmetric positive definite matrix whose largest eigenvalue is 1. We prove that for $0\le x\le \mathit{\epsilon }{n}^{1∕6}$,

$$\left|\frac{\mathbb{P}(|Q^{1∕2}W|>x)}{\mathbb{P}(|Q^{1∕2}Z|>x)}-1\right|\le C\left(\frac{1+x^5}{det(Q^{1∕2})n}+\frac{x^6}{n}\right)\text{for}d\ge 5$$

and

$$\left|\frac{\mathbb{P}(|Q^{1∕2}W|>x)}{\mathbb{P}(|Q^{1∕2}Z|>x)}-1\right|\le C\left(\frac{1+x^3}{det(Q^{1∕2})n^{\frac{d}{d+1}}}+\frac{x^6}{n}\right)\text{for}1\le d\le 4,$$

where ε and C are positive constants depending only on $d,t_0$, and $c_0$. This is a first extension of Cramér-type moderate deviation to the multivariate setting with a faster convergence rate than $1∕\sqrt{n}$. The range of $x=o(n^{1∕6})$ for the relative error to vanish and the dimension requirement $d\ge 5$ 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 $1∕\sqrt{n}$.