Multivariate Gaussian approximation for region-based stabilizing functionals under i.i.d. (binomial) sampling

Develop multivariate normal approximation bounds for functionals of binomial point processes that are sums of region-based stabilizing score functions under independent and identically distributed sampling. In particular, establish counterparts to multivariate second-order Poincaré inequalities (or alternative techniques) that yield quantitative multivariate Gaussian approximation for such region-based stabilizing functionals, analogous to what is available for Poisson point processes.

Background

The paper derives multivariate Gaussian approximation bounds for random-forest-type estimators by leveraging region-based stabilization and multivariate second-order Poincaré inequalities for Poisson functionals. These inequalities are central to the multivariate normal approximation results (Theorems d2d3 and dconvex).

For i.i.d. sampling (binomial point processes), analogous multivariate second-order Poincaré inequalities are not available. While univariate results can be derived via adaptations of existing methods, multivariate normal approximation for region-based stabilizing functionals under i.i.d. sampling currently lacks the necessary tools, motivating the stated open problem.

References

"According to , second-order Poincaré inequalities for the multivariate normal approximation of Poisson functionals have no available counterparts for binomial point processes. Thus, there are no immediate versions of multivariate (i.e., for m>1) normal approximations (i.e., analogs of Theorem~\ref{d2d3} and Theorem~\ref{dconvex}) of region-based stabilizing functionals under i.i.d.\ samples. This remains an open problem, with applications beyond the scope of the current work."

— Multivariate Gaussian Approximation for Random Forest via Region-based Stabilization  (2403.09960 - Shi et al., 2024) in Remark [Binomial Point Processes], Section 3 (Main results)

In fact, the optimal dimension dependence remains unclear even for sums of independent random vectors; we refer to the discussion following Theorem~1 of \citet{zhang2026wasserstein} for further details.

— Gaussian Approximation for Multivariate Martingale Sums from Uniformly Ergodic Markov Chains  (2609.09480 - Zhang et al., 8 Sep 2026) in Section 2, subsection “Dimension dependence”