Three-Parameter Approximations of Sums of Locally Dependent Random Variables via Stein's Method

TL;DR

This paper develops three-parameter approximations for sums of locally dependent non-negative integer variables using Stein's method, providing sharper error bounds and applications to classical combinatorial problems.

Contribution

It introduces a new three-parameter approximation framework with improved error bounds for sums of locally dependent variables using Stein's method.

Findings

Sharper upper error bounds than existing results.

Effective normal approximation for sums of dependent variables.

Applications to classical combinatorial problems like Erdős-Rényi graphs.

Abstract

Let $\{X_{i}, i\in J\}$ be a family of locally dependent non-negative integer-valued random variables with finite expectations and variances. We consider the sum $W=\sum_{i\in J}X_i$ and use Stein's method to establish general upper error bounds for the total variation distance $d_{TV}(W, M)$, where $M$ represents a three-parameter random variable. As a direct consequence, we obtain a discretized normal approximation for $W$. As applications, we study in detail four well-known examples, which are counting vertices of all edges point inward, birthday problem, counting monochromatic edges in uniformly colored graphs, and triangles in the Erd\H{o}s-R\'{e}nyi random graph. Through delicate analysis and computations, we obtain sharper upper error bounds than existing results.