Bootstrap Consistency for Quadratic Forms of Sample Averages with Increasing Dimension

This paper establishes consistency of the weighted bootstrap for quadratic forms $( n^{-1/2} \sum_{i=1}^{n} Z_{i,n})^{T}( n^{-1/2} \sum_{i=1}^{n} Z_{i,n})$ where $(Z_{i,n})_{i=1}^{n}$ are mean zero, independent $\mathbb{R}^{d}$-valued random variables and $d=d(n)$ is allowed to grow with the sample size $n$, slower than $n^{1/4}$. The proof relies on an adaptation of Lindeberg interpolation technique whereby we simplify the original problem to a Gaussian approximation problem.