High-dimensional CLT: Improvements, non-uniform extensions and large deviations

Central limit theorems (CLTs) for high-dimensional random vectors with dimension possibly growing the sample size have received a lot of attention in recent times. Chernozhukov et al. (Ann. Probab. 45 (2017) 2309–2352) proved Berry–Esseen type result averages class sparsely convex sets including hyperrectangles as special case and they that rate convergence can be upper bounded by $n^{-1/6}$ up to polynomial factor $\log p$ (where $n$ represents $p$ denotes dimension). Convergence zero bound requires $\log^{7}p=o(n)$. We improve upon their result, hyperrectangles, which only $\log^{4}p=o(n)$ (in best case). This improvement is made possible sharper dimension-free anti-concentration inequality Gaussian process on compact metric space. In addition, we prove two non-uniform variants CLT based large deviation results variables Banach space Bentkus, Rackauskas, Paulauskas. apply our context post-selection inference linear regression empirical processes.