Cram\'er-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 $\mathbb{E} e^{t_0|X_1|}\leq c_0<\infty$, where $|\cdot|$ denotes the Euclidean norm. Let $W=\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$ 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 $0\leq x\leq \varepsilon n^{1/6}$, \begin{equation*} \left| \frac{\mathbb{P}(|Q^{1/2}W|>x)}{\mathbb{P}(|Q^{1/2}Z|>x)}-1 \right|\leq C \left( \frac{1+x^5}{\det{(Q^{1/2})}n}+\frac{x^6}{n}\right) \quad \text{for}\ d\geq 5 \end{equation*} and \begin{equation*} \left| \frac{\mathbb{P}(|Q^{1/2}W|>x)}{\mathbb{P}(|Q^{1/2}Z|>x)}-1 \right|\leq C \left( \frac{1+x^3}{\det{(Q^{1/2})}n^{\frac{d}{d+1}}}+\frac{x^6}{n}\right) \quad \text{for}\ 1\leq d\leq 4, \end{equation*} where $\varepsilon$ and $C$ are positive constants depending only on $d, t_0$, and $c_0$. This is a first extension of Cram\'er-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 $d\geq 5$ 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 $1/\sqrt{n}$.