🤖 AI Summary
This paper addresses the challenge of applying Yurinskii coupling under ℓₚ-norms (1 ≤ p ≤ ∞) to weakly approximable martingales. Methodologically, it introduces three innovations: (1) the first non-asymptotic ℓₚ-norm Yurinskii coupling for approximate martingales, substantially relaxing classical assumptions on independence, moment conditions, and exponential decay; (2) extension of coupling variables to generalized Gaussian mixtures, improving model flexibility; and (3) a novel third-order coupling scheme that sharply tightens approximation error bounds. The analysis integrates tools from mixing martingale theory, high-dimensional central limit theorems, empirical process theory, and local polynomial modeling. Key contributions include a central limit theorem for high-dimensional martingale vectors, a uniform strong Gaussian-mixture approximation for martingale empirical processes, and rigorous inferential guarantees—particularly for constructing confidence bands in nonparametric regression.
📝 Abstract
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.