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

Local limit theorems via Landau–Kolmogorov inequalities

Adrian Röllin and Nathan Ross

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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.

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

First available in Project Euclid: 21 April 2015

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


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

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