Gaussian fluctuations of generalized $U$-statistics and subgraph counting in the binomial random-connection model

TL;DR

This paper establishes normal approximation bounds for generalized U-statistics involving complex dependencies, with applications to counting subgraphs in a generalized random-connection model, extending classical Erdős-Rényi results.

Contribution

It introduces a cumulant-based method for deriving normal approximation bounds for generalized U-statistics with dependent structures.

Findings

Derived Kolmogorov distance bounds for generalized U-statistics

Established moderate deviation principles for these statistics

Applied results to subgraph counting in a binomial random-connection model

Abstract

We derive normal approximation bounds for generalized $U$-statistics of the form \begin{equation*} S_{n,k}(f):=\sum_{ 1 \leq \beta (1),\dots,\beta (k) \leq n \atop \beta (i)\ne\beta (j), \ 1\leq i\ne j \leq k} f\big(X_{\beta (1)},\dots,X_{\beta (k)},Y_{\beta (1),\beta (2)},\dots,Y_{\beta (k-1),\beta (k)}\big), \end{equation*} where $\{X_i\}_{i=1}^n$ and $\{Y_{i,j}\}_{1\le i<j\le n}$ are independent sequences of i.i.d. random variables. Our approach relies on moment identities and cumulant bounds that are derived using partition diagram arguments. Normal approximation bounds in the Kolmogorov distance and moderate deviation results are then obtained by the cumulant method. Those results are applied to subgraph counting in the binomial random-connection model, which is a generalization of the Erd\H{o}s-R\'enyi model.