## 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

#### 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
Revised: June 2019
First available in Project Euclid: 26 November 2019

https://projecteuclid.org/euclid.bj/1574758839

Digital Object Identifier
doi:10.3150/19-BEJ1141

Mathematical Reviews number (MathSciNet)
MR4036045

Zentralblatt MATH identifier
07140510

#### 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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