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2019 Quantitative CLTs for symmetric $U$-statistics using contractions
Christian Döbler, Giovanni Peccati
Electron. J. Probab. 24: 1-43 (2019). DOI: 10.1214/19-EJP264

Abstract

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).

Citation

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Christian Döbler. Giovanni Peccati. "Quantitative CLTs for symmetric $U$-statistics using contractions." Electron. J. Probab. 24 1 - 43, 2019. https://doi.org/10.1214/19-EJP264

Information

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

zbMATH: 07021646
MathSciNet: MR3916325
Digital Object Identifier: 10.1214/19-EJP264

Subjects:
Primary: 60D05, 60F05, 62G99

JOURNAL ARTICLE
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Vol.24 • 2019
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