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Quantitative CLTs for symmetric \(U\)-statistics using contractions. (English) Zbl 1442.60034
Summary: We consider sequences of symmetric \(U\)-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of contraction operators. Our results represent an explicit counterpart to analogous criteria that are available for sequences of random variables living on the Gaussian, Poisson or Rademacher chaoses, and are perfectly tailored for geometric applications. As a demonstration of this fact, we develop explicit bounds for subgraph counting in generalised random graphs on Euclidean spaces; special attention is devoted to the so-called ’dense parameter regime’ for uniformly distributed points, for which we deduce CLTs that are new even in their qualitative statement, and that substantially extend classical findings by S. R. Jammalamadaka and S. Janson [Ann. Probab. 14, 1347–1358 (1986; Zbl 0604.60023)] and R. N. Bhattacharya and J. K. Ghosh [J. Multivariate Anal. 43, No. 2, 300–330 (1992; Zbl 0764.60025)].
60F05 Central limit and other weak theorems
60D05 Geometric probability and stochastic geometry
62G99 Nonparametric inference
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