Skewness correction in tail probability approximations for sums of local statistics

TL;DR

This paper enhances tail probability approximations for sums of local statistics by correcting skewness, applying advanced probabilistic techniques to improve accuracy in various combinatorial and graph models.

Contribution

It extends skewness correction methods to local statistics, including U-statistics and subgraph counts, using Stein's method and concentration inequalities.

Findings

Improved tail probability approximations for local statistics.

Application to k-runs, U-statistics, and Erdős-Rényi graph counts.

Development of exponential concentration inequalities and moderate deviations.

Abstract

Correcting for skewness can result in more accurate tail probability approximations in the central limit theorem for sums of independent random variables. In this paper, we extend the theory to sums of local statistics of independent random variables and apply the result to $k$-runs, U-statistics, and subgraph counts in the Erd\"os-R\'enyi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cram\'er-type moderate deviations via Stein's method.