Yurinskii’s coupling for martingales
Matias D. Cattaneo, Ricardo P. Masini, William G. Underwood
The Annals of Statistics · 2025 · Volume 53 · Issue 5
Abstract
Yurinskii’s coupling is a popular theoretical tool for nonasymptotic 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 ℓ2-norm for sums of independent random vectors, it has recently been extended both to the ℓp-norm, for 1≤p≤∞, and to vector-valued martingales in ℓ2-norm, under some strong conditions. We present as our main result a Yurinskii coupling for approximate martingales in ℓp-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.
Keywords
60G42, 62E20, 62G20, central limit theorems, coupling, dependent data, Empirical processes, Gaussian mixture approximation, Local polynomial estimation, Martingales, mixingales, series estimation, strong approximation, time series, uniform inference
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