Bernoulli

  • Bernoulli
  • Volume 25, Number 2 (2019), 1076-1104.

Error bounds in local limit theorems using Stein’s method

A.D. Barbour, Adrian Röllin, and Nathan Ross

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Abstract

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.

Article information

Source
Bernoulli, Volume 25, Number 2 (2019), 1076-1104.

Dates
Received: July 2017
Revised: November 2017
First available in Project Euclid: 6 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1551862844

Digital Object Identifier
doi:10.3150/17-BEJ1013

Mathematical Reviews number (MathSciNet)
MR3920366

Zentralblatt MATH identifier
07049400

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

Citation

Barbour, A.D.; Röllin, Adrian; Ross, Nathan. Error bounds in local limit theorems using Stein’s method. Bernoulli 25 (2019), no. 2, 1076--1104. doi:10.3150/17-BEJ1013. https://projecteuclid.org/euclid.bj/1551862844


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