Three-parameter approximations of sums of locally dependent random variables via Stein’s method

Let {X_i, i ∈ J} be a family of locally dependent non-negative integer-valued random variables with finite expectations and variances. We consider the sum W = ∑_i∈J X_i and apply 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 four well-known examples in detail: counting vertices where all edges point inward, the birthday problem, counting monochromatic edges in uniformly colored graphs, and triangles in the Erdős-Rényi random graph. Through delicate analysis and computation, we obtain sharper upper error bounds than existing results.