## The Annals of Applied Probability

### Central limit theorems in the configuration model

#### Abstract

We prove a general normal approximation theorem for local graph statistics in the configuration model, together with an explicit bound on the error in the approximation with respect to the Wasserstein metric. Such statistics take the form $T:=\sum_{v\in V}H_{v}$, where $V$ is the vertex set, and $H_{v}$ depends on a neighbourhood in the graph around $v$ of size at most $\ell$. The error bound is expressed in terms of $\ell$, $|V|$, an almost sure bound on $H_{v}$, the maximum vertex degree $d_{\max}$ and the variance of $T$. Under suitable assumptions on the convergence of the empirical degree distributions to a limiting distribution, we deduce that the size of the giant component in the configuration model has asymptotically Gaussian fluctuations.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 2 (2019), 1046-1069.

Dates
Revised: July 2018
First available in Project Euclid: 24 January 2019

https://projecteuclid.org/euclid.aoap/1548298936

Digital Object Identifier
doi:10.1214/18-AAP1425

Mathematical Reviews number (MathSciNet)
MR3910023

Subjects
Primary: 60F05: Central limit and other weak theorems

#### Citation

Barbour, A. D.; Röllin, Adrian. Central limit theorems in the configuration model. Ann. Appl. Probab. 29 (2019), no. 2, 1046--1069. doi:10.1214/18-AAP1425. https://projecteuclid.org/euclid.aoap/1548298936

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