Electronic Journal of Probability

Quantitative CLTs for symmetric $U$-statistics using contractions

Christian Döbler and Giovanni Peccati

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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 Jammalamadaka and Janson (1986) and Bhattacharaya and Ghosh (1992).

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 5, 43 pp.

Received: 7 February 2018
Accepted: 5 January 2019
First available in Project Euclid: 9 February 2019

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Primary: 60F05: Central limit and other weak theorems 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 62G99: None of the above, but in this section

$U$-statistics central limit theorem error bounds contractions product formula random geometric graphs Hoeffding decomposition Stein’s method exchangeable pairs

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Döbler, Christian; Peccati, Giovanni. Quantitative CLTs for symmetric $U$-statistics using contractions. Electron. J. Probab. 24 (2019), paper no. 5, 43 pp. doi:10.1214/19-EJP264. https://projecteuclid.org/euclid.ejp/1549681361

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