Normal Approximation for U-Statistics with Cross-Sectional Dependence

Weiguang Liu

Papers from arXiv.org

Abstract: We establish normal approximation in the Wasserstein metric and central limit theorems for both non-degenerate and degenerate U-statistics with cross-sectionally dependent samples using Stein's method. For the non-degenerate case, our results extend recent studies on the asymptotic properties of sums of cross-sectionally dependent random variables. The degenerate case is more challenging due to the additional dependence induced by the nonlinearity of the U-statistic kernel. Through a specific implementation of Stein's method, we derive convergence rates under conditions on the mixing rate, the sparsity of the cross-sectional dependence structure, and the moments of the U-statistic kernel. Finally, we demonstrate the application of our theoretical results with a nonparametric specification test for data with cross-sectional dependence.

Date: 2024-11, Revised 2025-03
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