论文标题
Yurinskii的Martingales耦合
Yurinskii's Coupling for Martingales
论文作者
论文摘要
Yurinskii的耦合是一种流行的理论工具，用于数学统计和应用概率中的非反应分布分析，提供高斯强近似，并在易于证实的条件下具有明显的误差。最初以$ \ ell_2 $ -norm表示独立随机向量的总和，最近已将其扩展到$ \ ell_p $ -norm，价格为$ 1 \ leq p \ leq p \ leq \ infty $，在$ \ ell_2 $ norm的情况下，在$ \ ell_2 $的条件下，在$ \ ell_2 $ norm中，它以$ 1 \ ell_p $ -norm的价格扩展到了vector-vector-valite-valued martingales。我们作为我们的主要结果，在$ \ ell_p $ -norm中，yurinskii耦合在比以前施加的条件下要弱得多。我们的配方进一步允许耦合变量遵循更一般的高斯混合物分布，我们提供了一种新颖的三阶耦合方法，该方法在某些设置中可以更近似。我们将主要结果专门用于混合群，蛋黄酱和独立的数据，并得出均匀的高斯混合物，以实现Martingale经验过程。还提供了基于非参数分区的和局部多项式回归程序的应用，以及中央限制定理的高维martingale矢量。
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 verifiable 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, alongside central limit theorems for high-dimensional martingale vectors.
