{"ID":23507328,"CreatedAt":"2026-09-18T02:21:44.056544415Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20529","arxiv_id":"2609.20529","title":"Approximation Theorems for High-Dimensional Canonical U-Statistics: Gaussian Chaos and Phase Transition","abstract":"We study simultaneous inference for maxima of canonical order-two $U$-statistics in high dimension. Degeneracy makes quadratic fluctuations leading, so ordinary Gaussian calibration can fail even after exact variance normalization. We show that the appropriate general target is a joint signed Gaussian quadratic chaos and establish a general approximation result that permits indefinite kernels. The general anti-concentration bound is too crude for high-dimensional inference, and we obtain sharper bounds under additional spectral structure. We also identify a phase transition from a non-Gaussian signed-chaos maximum to its covariance-matched Gaussian counterpart driven by the effective rank. For feasible inference, we propose a Gaussian multiplier bootstrap that avoid estimating eigensystems, and establish its validity. Two applications and extensive numerical simulations further illustrate the scope and practical performance of the proposed framework.","short_abstract":"We study simultaneous inference for maxima of canonical order-two $U$-statistics in high dimension. Degeneracy makes quadratic fluctuations leading, so ordinary Gaussian calibration can fail even after exact variance normalization. We show that the appropriate general target is a joint signed Gaussian quadratic chaos a...","url_abs":"https://arxiv.org/abs/2609.20529","url_pdf":"https://arxiv.org/pdf/2609.20529v1","authors":"[\"Leheng Cai\",\"Qirui Hu\"]","published":"2026-09-17T15:05:16Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}