Refined Berry-Esseen bounds under local dependence

In this paper, we establish Berry–Esseen bounds for both self-normalized and non-self-normalized sums of locally dependent random variables. The proofs are based on Stein’s method together with a concentration inequality approach. We develop a new class of concentration inequalities that extend classical results and achieve optimal convergence rates under more general dependence structures. As applications, we apply our main results to derive sharper Berry–Esseen bounds for graph dependency, distributed $U$-statistics, constrained $U$-statistics, and decorated injective homomorphism sums.