Strong Gaussian approximation for U-statistics in high dimensions and beyond

TL;DR

This paper develops a strong Gaussian approximation for high-dimensional U-statistics, enabling better understanding and inference in complex statistical models with diverging dimensions.

Contribution

It introduces a novel coupling technique for high-dimensional U-statistics, providing explicit error bounds and functional Gaussian limits without relying on bootstrap methods.

Findings

Explicit approximation error bounds are established.

Applications include change-point detection and pivotal testing procedures.

Framework accommodates heavy-tailed distributions and bounded kernels.

Abstract

We establish a strong Gaussian approximation for high-dimensional non-degenerate U-statistics with diverging dimension. Under mild assumptions, we construct, on a sufficiently rich probability space, a Gaussian process that uniformly approximates the entire sequential U-statistic process. The approximation error is explicitly characterized and vanishes under polynomial growth of the dimension. The key technical contribution is a sharp martingale maximal inequality for completely degenerate U-statistics, combined with a high-dimensional strong approximation for independent sums. This coupling yields functional Gaussian limits without relying on $\mathcal{L}^\infty$-type bounds or bootstrap arguments. The theory is illustrated through three representative examples of U-statistics: the spatial Kendall's tau matrix, the multivariate Gini's mean difference, and the characteristic dispersion parameter. As applications, we derive Brownian bridge approximations for U-statistic-based change-point statistics and develop a self-normalized relevant testing procedure whose limiting distribution is fully pivotal. The framework naturally accommodates bounded kernels and therefore remains valid under heavy-tailed distributions. Overall, our results provide a unified probability-theoretic foundation for high-dimensional inference based on U-statistics.