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

Weijia Li, Leheng Cai, Qirui Hu

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.

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

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 -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.
Paper Structure (9 sections, 10 theorems, 40 equations)

This paper contains 9 sections, 10 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Sequential Gaussian approximation for partial sums
  5. Gaussian approximation for full sums
  6. Statistical applications
  7. Application to relevant tests
  8. Application to change point analysis
  9. Concluding remark

Key Result

Theorem 1

Under Assumptions (A1)-(A2), there exists a sequence of independent Gaussian random vectors $\{\bm Z_i\}_{i=1}^n$ with $\bm Z_i \sim \mathcal{N}(\bm{0}, \bm{\Sigma})$ such that, for the Gaussian partial sum process $\bm{W}_k = \sum_{i=1}^k \bm Z_i/\sqrt{n}$, Under (A3), the above approximation error vanishes asymptotically.

Theorems & Definitions (17)

  • Example 1: Multivariate Gini's Mean Difference
  • Example 2: Characteristic dispersion parameter
  • Example 3: Spatial Kendall's tau matrix
  • Theorem 1
  • Proposition 1
  • Lemma 2.1: Maximal Inequality for Degenerate U-statistics
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Example \ref{exa:GMD}: Multivariate Gini's Mean Difference (continued)
  • ...and 7 more