Yurinskii's coupling is a popular theoretical tool for non-asymptotic distributional analysis in mathematical statistics and applied probability, offering a Gaussian strong approximation with an explicit error bound under easily verified conditions. Originally stated in $\ell^2$-norm for sums of independent random vectors, it has recently been extended both to the $\ell^p$-norm, for $1 \leq p \leq \infty$, and to vector-valued martingales in $\ell^2$-norm, under some strong conditions. We present as our main result a Yurinskii coupling for approximate martingales in $\ell^p$-norm, under substantially weaker conditions than those previously imposed. Our formulation further allows for the coupling variable to follow a more general Gaussian mixture distribution, and we provide a novel third-order coupling method which gives tighter approximations in certain settings. We specialize our main result to mixingales, martingales, and independent data, and derive uniform Gaussian mixture strong approximations for martingale empirical processes. Applications to nonparametric partitioning-based and local polynomial regression procedures are provided.

翻译：Yurinskii耦合是数理统计与应用概率中非渐近分布分析的重要理论工具，可在易于验证的条件下提供具有显式误差界的高斯强近似。该结论最初针对独立随机向量和的ℓ²范数提出，近年来虽已推广至ℓ^p范数（1 ≤ p ≤ ∞）及在ℓ²范数下带强条件的向量值鞅，但现有结果强条件限制较多。本文主要结果在显著弱于既往条件的假设下，建立了ℓ^p范数下近似鞅的Yurinskii耦合。我们提出耦合变量可服从更一般的高斯混合分布，并首创三阶耦合方法，能在特定场景下实现更紧密的近似。将主要结论具体应用于混合序列、鞅及独立数据，导出鞅经验过程的高斯混合强一致收敛逼近。最后给出该方法在非参数划分与局部多项式回归中的应用实例。