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

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.