Open Access
May 2019 Error bounds in local limit theorems using Stein’s method
A.D. Barbour, Adrian Röllin, Nathan Ross
Bernoulli 25(2): 1076-1104 (May 2019). DOI: 10.3150/17-BEJ1013


We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in the context of the Hoeffding combinatorial central limit theorem with integer valued summands, of the number of isolated vertices in an Erdős–Rényi random graph, and of the Curie–Weiss model of magnetism, where we provide optimal or near optimal rates of convergence in the local limit metric. In the Hoeffding example, even the discrete normal approximation bounds seem to be new. The general result follows from Stein’s method, and requires a new bound on the Stein solution for the Poisson distribution, which is of general interest.


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A.D. Barbour. Adrian Röllin. Nathan Ross. "Error bounds in local limit theorems using Stein’s method." Bernoulli 25 (2) 1076 - 1104, May 2019.


Received: 1 July 2017; Revised: 1 November 2017; Published: May 2019
First available in Project Euclid: 6 March 2019

zbMATH: 07049400
MathSciNet: MR3920366
Digital Object Identifier: 10.3150/17-BEJ1013

Keywords: Approximation error , Curie–Weiss model , Erdős–Rényi random graph , Hoeffding combinatorial statistic , local limit theorem

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 2 • May 2019
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