[Paper Review] Wasserstein-2 bounds in normal approximation under local dependence

Xiao Fang|arXiv (Cornell University)|Jul 16, 2018

Geometric Analysis and Curvature Flows9 references4 citations

TL;DR

This paper establishes a general Wasserstein-2 distance bound in normal approximation for sums of locally dependent random variables using Stein's method and asymptotic expansions of second-order differentiable functions. The key result provides explicit bounds in terms of third- and fourth-order moments and neighborhood dependence structures, with applications to m-dependent sequences, U-statistics, and subgraph counts in Erdős-Rényi random graphs, along with a conjecture on p-Wasserstein bounds for general p.

ABSTRACT

We obtain a general bound for the Wasserstein-2 distance in normal approximation for sums of locally dependent random variables. The proof is based on an asymptotic expansion for expectations of second-order differentiable functions of the sum. We apply the main result to obtain Wasserstein-2 bounds in normal approximation for sums of $m$-dependent random variables, U-statistics and subgraph counts in the Erdős-Rényi random graph. We state a conjecture on Wasserstein-$p$ bounds for any positive integer $p$ and provide supporting arguments for the conjecture.

Motivation & Objective

• To derive a general Wasserstein-2 distance bound in normal approximation for sums of locally dependent random variables.
• To extend existing results on Wasserstein-1 and Kolmogorov distance bounds to the Wasserstein-2 metric under local dependence.
• To provide explicit bounds involving third- and fourth-order moments and neighborhood dependence structures.
• To apply the main result to m-dependent sequences, U-statistics, and subgraph counts in Erdős-Rényi random graphs.
• To propose and support a conjecture on p-Wasserstein bounds for any positive integer p.

Proposed method

• Use Stein's method to derive an asymptotic expansion for expectations of second-order differentiable functions of the sum W.
• Apply Zolotarev’s ideal distance of order 2 to control the Wasserstein-2 distance between W and a sum of i.i.d. random variables.
• Leverage the triangle inequality and known Wasserstein-2 bounds for i.i.d. sums to bound the distance between W and the standard normal distribution.
• Define local dependence via three conditions (LD1)–(LD3) involving shrinking neighborhoods A_i, A_ij, A_ijk for each X_i.
• Use cumulant expansions up to order 4 to control error terms in the asymptotic expansion.
• Support the conjecture on p-Wasserstein bounds via analogous asymptotic expansions and Young’s inequality for higher-order cumulants.

Experimental results

Research questions

• RQ1 What is the optimal Wasserstein-2 distance bound in normal approximation for sums of locally dependent random variables?
• RQ2 How can Stein’s method and asymptotic expansions be adapted to bound the Wasserstein-2 distance under local dependence?
• RQ3 Can the approach used for p=2 be generalized to derive bounds for p-Wasserstein distances for any positive integer p?
• RQ4 What are the explicit moment-based bounds for Wasserstein-2 distance in m-dependent sequences, U-statistics, and subgraph counts in Erdős-Rényi random graphs?
• RQ5 What conditions on the dependence structure and moments ensure tight Wasserstein-2 bounds in normal approximation?

Key findings

• The paper establishes a Wasserstein-2 bound of the form $\mathcal{W}_2(\mathcal{L}(W), N(0,1)) \leq C[|\beta| + (\gamma_1 + \gamma_2 + \gamma_3)^{1/2}]$, where $\beta$ captures third-order dependencies and $\gamma_i$ quantify higher-order moment contributions.
• The bound applies to m-dependent sequences, U-statistics, and subgraph counts in Erdős-Rényi random graphs, providing explicit rates in terms of dependence neighborhoods and moments.
• The proof technique combines Stein’s method with asymptotic expansions of second-order differentiable functions and Zolotarev’s ideal distance of order 2.
• The authors provide supporting arguments for a conjecture that similar bounds hold for all positive integers p, based on higher-order cumulant expansions and Young’s inequality.
• The method is robust to weak dependence, as demonstrated by the use of shrinking neighborhoods A_i, A_ij, A_ijk in the local dependence structure.
• The result generalizes prior Wasserstein-1 bounds by Barbour, Karoński, and Ruciński (1989) and extends the scope of normal approximation to non-i.i.d. settings with local dependence.