Electronic Journal of Probability
- Electron. J. Probab.
- Volume 24 (2019), paper no. 5, 43 pp.
Quantitative CLTs for symmetric $U$-statistics using contractions
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).
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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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