Bernoulli

  • Bernoulli
  • Volume 26, Number 1 (2020), 587-615.

Normal approximation for sums of weighted $U$-statistics – application to Kolmogorov bounds in random subgraph counting

Nicolas Privault and Grzegorz Serafin

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Abstract

We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and weighted $U$-statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized subgraph counts in the Erdős–Rényi random graph. This approach completely solves a long-standing conjecture in the general setting of arbitrary graph counting, while recovering recent results obtained for triangles and improving other bounds in the Wasserstein distance.

Article information

Source
Bernoulli, Volume 26, Number 1 (2020), 587-615.

Dates
Received: October 2018
Revised: June 2019
First available in Project Euclid: 26 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1574758839

Digital Object Identifier
doi:10.3150/19-BEJ1141

Mathematical Reviews number (MathSciNet)
MR4036045

Zentralblatt MATH identifier
07140510

Keywords
Berry–Esseen bound central limit theorem Kolmogorov distance Malliavin–Stein method normal approximation random graph Stein–Chen method subgraph count

Citation

Privault, Nicolas; Serafin, Grzegorz. Normal approximation for sums of weighted $U$-statistics – application to Kolmogorov bounds in random subgraph counting. Bernoulli 26 (2020), no. 1, 587--615. doi:10.3150/19-BEJ1141. https://projecteuclid.org/euclid.bj/1574758839


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