Kolmogorov bounds for decomposable random variables and subgraph counting by the Stein-Tikhomirov method

TL;DR

This paper discusses the Stein-Tikhomirov method, combining Stein's approach with characteristic functions, to derive bounds on convergence rates in normal approximation problems, including subgraph counting.

Contribution

It extends the Stein-Tikhomirov method to new contexts, providing Kolmogorov bounds for decomposable variables and subgraph counts in random graphs.

Findings

Established Kolmogorov bounds for decomposable random variables.

Applied the method to subgraph counting in Erdős-Rényi graphs.

Improved convergence rate estimates in normal approximation.

Abstract

In his work \cite{Ti80}, Tikhomirov combined elements of Stein's method with the theory of characteristic functions to derive Kolmogorov bounds for the convergence rate in the central limit theorem for a normalized sum of a stationary sequence of random variables satisfying one of several weak dependency conditions. The combination of elements of Stein's method with the theory of characteristic functions is sometimes called \emph{Stein-Tikhomirov method}. \citet*{AMPS17} successfully used the Stein-Tikhomirov method to bound the convergence rate in contexts with non-Gaussian targets. \citet*{Ro17} used the Stein-Tikhomirov method to bound the convergence rate in the Kolmogorov distance for normal approximation of normalized triangle counts in the Erd\"os-R\'enyi random graph.