## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 4 (2017), 1992-2031.

### New Berry–Esseen bounds for functionals of binomial point processes

Raphaël Lachièze-Rey and Giovanni Peccati

#### Abstract

We obtain explicit Berry–Esseen bounds in the Kolmogorov distance for the normal approximation of nonlinear functionals of vectors of independent random variables. Our results are based on the use of Stein’s method and of random difference operators, and generalise the bounds obtained by Chatterjee (2008), concerning normal approximations in the Wasserstein distance. In order to obtain lower bounds for variances, we also revisit the classical Hoeffding decompositions, for which we provide a new proof and a new representation. Several applications are discussed in detail: in particular, new Berry–Esseen bounds are obtained for set approximations with random tessellations, as well as for functionals of coverage processes.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 4 (2017), 1992-2031.

**Dates**

Received: May 2015

Revised: January 2016

First available in Project Euclid: 30 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1504080024

**Digital Object Identifier**

doi:10.1214/16-AAP1218

**Mathematical Reviews number (MathSciNet)**

MR3693518

**Zentralblatt MATH identifier**

1374.60023

**Subjects**

Primary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

**Keywords**

Berry–Esseen bounds binomial processes covering processes random tessellations stochastic geometry Stein’s method

#### Citation

Lachièze-Rey, Raphaël; Peccati, Giovanni. New Berry–Esseen bounds for functionals of binomial point processes. Ann. Appl. Probab. 27 (2017), no. 4, 1992--2031. doi:10.1214/16-AAP1218. https://projecteuclid.org/euclid.aoap/1504080024