Bernoulli

  • Bernoulli
  • Volume 21, Number 2 (2015), 851-880.

Local limit theorems via Landau–Kolmogorov inequalities

Adrian Röllin and Nathan Ross

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Abstract

In this article, we prove new inequalities between some common probability metrics. Using these inequalities, we obtain novel local limit theorems for the magnetization in the Curie–Weiss model at high temperature, the number of triangles and isolated vertices in Erdős–Rényi random graphs, as well as the independence number in a geometric random graph. We also give upper bounds on the rates of convergence for these local limit theorems and also for some other probability metrics. Our proofs are based on the Landau–Kolmogorov inequalities and new smoothing techniques.

Article information

Source
Bernoulli Volume 21, Number 2 (2015), 851-880.

Dates
First available in Project Euclid: 21 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1429624963

Digital Object Identifier
doi:10.3150/13-BEJ590

Mathematical Reviews number (MathSciNet)
MR3338649

Zentralblatt MATH identifier
1320.60065

Keywords
Curie–Weiss model Erdős–Rényi random graph Kolmogorov metric Landau–Kolmogorov inequalities local limit metric total variation metric Wasserstein metric

Citation

Röllin, Adrian; Ross, Nathan. Local limit theorems via Landau–Kolmogorov inequalities. Bernoulli 21 (2015), no. 2, 851--880. doi:10.3150/13-BEJ590. https://projecteuclid.org/euclid.bj/1429624963.


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