Kolmogorov bounds for the normal approximation of the number of triangles in the Erdos-Renyi random graph

TL;DR

This paper establishes optimal Kolmogorov bounds for the normal approximation of triangle counts in Erdős-Rényi graphs, using a novel Stein-Tikhomirov method, resolving a longstanding open problem.

Contribution

It provides the first matching Kolmogorov bounds for triangle counts, advancing the understanding of normal approximation in random graphs.

Findings

Bounds match the best Wasserstein bounds from prior work

Introduces a new Stein-Tikhomirov method variant

Resolves a long-standing open problem in the field

Abstract

We bound the error for the normal approximation of the number of triangles in the Erdos-Renyi random graph with respect to the Kolmogorov metric. Our bounds match the best available Wasserstein-bounds obtained by Barbour, Karonski and Rucinski (1989), resolving a long-standing open problem. The proofs are based on a new variant of the Stein-Tikhomirov method - a combination of Stein's method and characteristic functions introduced by Tikhomirov (1980).