The Annals of Probability

Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs and percolation

Kai Krokowski, Anselm Reichenbachs, and Christoph Thäle

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Abstract

A new Berry–Esseen bound for nonlinear functionals of nonsymmetric and nonhomogeneous infinite Rademacher sequences is established. It is based on a discrete version of the Malliavin–Stein method and an analysis of the discrete Ornstein–Uhlenbeck semigroup. The result is applied to sub-graph counts and to the number of vertices having a prescribed degree in the Erdős–Rényi random graph. A further application deals with a percolation problem on trees.

Article information

Source
Ann. Probab. Volume 45, Number 2 (2017), 1071-1109.

Dates
Received: March 2015
Revised: November 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947314

Digital Object Identifier
doi:10.1214/15-AOP1081

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60F05: Central limit and other weak theorems 60H07: Stochastic calculus of variations and the Malliavin calculus 82B43: Percolation [See also 60K35]

Keywords
Berry–Esseen bound central limit theorem Malliavin–Stein method Mehler’s formula percolation Rademacher functional random graph sub-graph count tree

Citation

Krokowski, Kai; Reichenbachs, Anselm; Thäle, Christoph. Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs and percolation. Ann. Probab. 45 (2017), no. 2, 1071--1109. doi:10.1214/15-AOP1081. https://projecteuclid.org/euclid.aop/1490947314.


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