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.

💡 Research Summary

This paper establishes refined Berry‑Esseen bounds for both non‑self‑normalized and self‑normalized sums of random variables that satisfy a local dependence structure. The authors introduce two conditions, (LD1) and (LD2). (LD1) requires that each variable X_i be independent of all variables outside a designated neighbor set A_i. (LD2) strengthens this by demanding that for any i and any j in A_i there exists a larger set A_{ij} containing A_i such that the pair {X_i, X_j} is independent of all variables outside A_{ij}. Compared with the previously used (LD2′), the new (LD2) allows much smaller dependence neighborhoods, which is crucial for obtaining sharper rates in many applications.

The main theoretical contributions are two theorems. Theorem 2.1 gives a Kolmogorov‑distance bound for the standardized sum W₁ = S/σ, where S = Σ_{i=1}^n X_i and σ² = Var(S). The bound involves two structural parameters: κ = max{ sup_i |N_i|, sup_{i,j} |A_{ij}| } and τ = sup_i |D_i|, where N_i and D_i encode the first‑order and second‑order dependence neighborhoods. The error is bounded by C κ² σ⁻³ Σ ‖X_i‖₃⁴ + C κ^{1/2}(κ+τ^{1/2}) σ⁻² ( Σ ‖X_i‖₄⁴ )^{1/2}. Only third and fourth moments are required, which is weaker than many earlier results that need higher moments.

Theorem 2.2 treats the self‑normalized statistic W₂ = S/V, where V is a data‑driven variance estimator built from the local sums Y_i = Σ_{j∈A_i} X_j. The bound has the same structure as Theorem 2.1 but is multiplied by a factor λ = κ Σ E X_i² / σ². In most practical settings λ is O(1), so the rate remains optimal.

A key methodological innovation is a new class of randomized concentration inequalities tailored to the (LD1)–(LD2) framework. These inequalities are derived via a recursive argument that extends the ideas of Chen and Shao (2004) and Chen, Röllin, and Xia (2021). They allow the authors to control the Stein equation solutions without invoking high‑order moment assumptions or complex martingale decompositions. Consequently, the constants in the Berry‑Esseen bounds are universal absolute constants.

The paper then applies the general theorems to four important statistical models:

1. Graph Dependency – When the random variables are indexed by vertices of a dependency graph with maximal degree d, the parameters κ and τ reduce to functions of d. The resulting bounds (3.1)–(3.2) match the Wasserstein‑1 bounds of Ross (2011) and improve upon earlier Berry‑Esseen bounds that scale as d⁵ or involve stronger moment conditions.

2. Distributed U‑Statistics – By partitioning a large sample into k blocks and averaging the block‑wise U‑statistics, the authors treat the resulting statistic as a sum of locally dependent variables. Under mild moment assumptions (finite fourth moment of the kernel) and block size conditions (k = O(N^α) with α < ½), Theorem 3.4 yields a Berry‑Esseen error of order O(N^{-1/2}), improving on previous work that required k = o(N) and additional regularity conditions.

3. Constrained U‑Statistics for m‑Dependent Sequences – For stationary m‑dependent sequences and constraints on index gaps, the authors define a constrained U‑statistic. By mapping the constraints into the (LD1)–(LD2) framework, they obtain bounds that depend polynomially on the dependence length m and the constraint vector, again with only third and fourth moments required.

4. Decorated Injective Homomorphism Sums – In combinatorial settings where one sums over injective graph homomorphisms with vertex decorations, the dependence structure is governed by the underlying graph’s maximal degree. The new bounds provide sharper convergence rates than existing literature, especially when the decoration introduces additional local interactions.

Overall, the paper delivers a unified, moment‑light approach to normal approximation under local dependence. The concentration‑inequality technique combined with Stein’s method yields optimal convergence rates (up to constants) for a broad class of dependent structures, and the applications demonstrate tangible improvements over prior results. The authors suggest future extensions to multivariate self‑normalized statistics, time‑series dependence, and higher‑order U‑statistics.