Yurinskii's Coupling for Martingales

Yurinskii's coupling is a popular tool for finite-sample distributional approximation in mathematical statistics and applied probability, offering a Gaussian strong approximation for sums of random vectors under easily verified conditions with an explicit rate of approximation. Originally stated for sums of independent random vectors in $\ell^2$-norm, it has recently been extended to the $\ell^p$-norm, where $1 \leq p \leq \infty$, and to vector-valued martingales in $\ell^2$-norm under some rather strong conditions. We provide as our main result a generalization of all of the previous forms of Yurinskii's coupling, giving a Gaussian strong approximation for martingales in $\ell^p$-norm under relatively weak conditions. We apply this result to some areas of statistical theory, including high-dimensional martingale central limit theorems and uniform strong approximations for martingale empirical processes. Finally we give a few illustrative examples in statistical methodology, applying our results to partitioning-based series estimators for nonparametric regression, distributional approximation of $\ell^p$-norms of high-dimensional martingales, and local polynomial regression estimators. We address issues of feasibility, demonstrating implementable statistical inference procedures in each section.