Paper
Strong Gaussian approximation for U-statistics in high dimensions and beyond
Authors
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.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.10595v1</id>\n <title>Strong Gaussian approximation for U-statistics in high dimensions and beyond</title>\n <updated>2026-03-11T09:51:14Z</updated>\n <link href='https://arxiv.org/abs/2603.10595v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.10595v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.ST'/>\n <category scheme='http://arxiv.org/schemas/atom' term='stat.ME'/>\n <published>2026-03-11T09:51:14Z</published>\n <arxiv:primary_category term='math.ST'/>\n <author>\n <name>Weijia Li</name>\n </author>\n <author>\n <name>Leheng Cai</name>\n </author>\n <author>\n <name>Qirui Hu</name>\n </author>\n </entry>"
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