Functional second-order Gaussian Poincaré inequalities

2025/06/16 by Vidotto, Anna, Zheng, Guangqu · 2 citations
#35Q35 #60F15 #60H30 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.13571

Abstract

In this paper, we work in the framework of Hilbert-valued Wiener structures and derive a functional version of the second-order Gaussian Poincaré inequality that leads to abstract bounds for Gaussian process approximation in d₂ distance. Our abstract bounds are flexible and can be applied in various examples including functional Breuer-Major central limit theorems, shallow neural networks, and spatial statistics of SPDEs solutions.

Citations

• Quantitative CLTs in Deep Neural Networks
• Gaussian fluctuations for the wave equation under rough random perturbations
• A Quantitative Functional Central Limit Theorem for Shallow Neural Networks
• Non-asymptotic approximations of Gaussian neural networks via second-order Poincaré inequalities
• Spatial ergodicity for SPDEs via Poincaré-type inequalities
• A Central Limit Theorem for the stochastic heat equation
• A central limit theorem for the stochastic heat equation
• Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency
• Second order Poincaré inequalities and CLTs on Wiener space
• Stein's method on Wiener chaos
• Fluctuations of eigenvalues and second order Poincaré inequalities

Cited by

• The Fourth-Moment Theorem on Hilbert Spaces
• A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces

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