• 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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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.

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

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

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Zentralblatt MATH identifier

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


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.

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