[Paper Review] Existence of Stein Kernels under a Spectral Gap, and Discrepancy Bound

Thomas A. Courtade, Max Fathi+1 more|arXiv (Cornell University)|Mar 22, 2017

Spectral Theory in Mathematical Physics33 references4 citations

TL;DR

This paper establishes the existence of Stein kernels for probability measures on ℝᵈ satisfying a Poincaré inequality, using calculus of variations. It derives dimension-free bounds on Stein discrepancy and proves that Stein discrepancy is strictly decreasing along the central limit theorem and controls skewness, leading to optimal convergence rates in Wasserstein distance W₂ for multidimensional CLTs with dimension-dependent optimality.

ABSTRACT

We establish existence of Stein kernels for probability measures on $\mathbb{R}^d$ satisfying a Poincaré inequality, and obtain bounds on the Stein discrepancy of such measures. Applications to quantitative central limit theorems are discussed, including a new CLT in Wasserstein distance $W_2$ with optimal rate and dependence on the dimension. As a byproduct, we obtain a stability version of an estimate of the Poincaré constant of probability measures under a second moment constraint. The results extend more generally to the setting of converse weighted Poincaré inequalities. The proof is based on simple arguments of calculus of variations. Further, we establish two general properties enjoyed by the Stein discrepancy, holding whenever a Stein kernel exists: Stein discrepancy is strictly decreasing along the CLT, and it controls the skewness of a random vector.

Motivation & Objective

• To establish sufficient conditions for the existence of Stein kernels in arbitrary dimensions.
• To derive dimension-free bounds on Stein discrepancy for measures satisfying a Poincaré inequality.
• To show that Stein discrepancy is strictly decreasing along the central limit theorem (CLT) when a Stein kernel exists.
• To prove that Stein discrepancy controls the skewness of a random vector.
• To obtain optimal convergence rates in Wasserstein distance W₂ for the multidimensional central limit theorem.

Proposed method

• Uses calculus of variations to prove existence of Stein kernels under a Poincaré inequality.
• Applies the integration-by-parts formula (1) to define Stein kernels via the Hilbert-Schmidt inner product.
• Derives bounds on Stein discrepancy using the Poincaré constant and second moment, independent of dimension.
• Establishes monotonicity of Stein discrepancy along the CLT using properties of the Ornstein-Uhlenbeck semigroup.
• Uses the relation between Stein discrepancy and W₂ distance to derive stability estimates for the Poincaré constant.
• Extends results to converse weighted Poincaré inequalities via Hölder’s inequality and duality.

Experimental results

Research questions

• RQ1 Under what conditions does a Stein kernel exist for a probability measure on ℝᵈ?
• RQ2 Can Stein discrepancy be bounded in a dimension-free manner for measures satisfying a Poincaré inequality?
• RQ3 Does Stein discrepancy decrease monotonically along the central limit theorem when a Stein kernel exists?
• RQ4 Can Stein discrepancy be used to control the skewness of a random vector?
• RQ5 What is the optimal rate of convergence in Wasserstein distance W₂ for the multidimensional central limit theorem under a Poincaré inequality?

Key findings

• Stein kernels exist for any probability measure on ℝᵈ satisfying a Poincaré inequality, resolving an open question from [42].
• Stein discrepancy is strictly decreasing along the CLT, analogous to entropy and Fisher information.
• Stein discrepancy controls the skewness of a random vector, providing a quantitative measure of non-Gaussianity.
• The paper establishes a W₂ convergence rate of order O(1/n) for the multidimensional CLT, which is optimal and dimension-dependent.
• A stable lower bound on the Poincaré constant is derived: Cₚ ≥ 1 + W₂(ν, γ)²/d, under second moment normalization.
• The results extend to converse weighted Poincaré inequalities, with a generalized bound involving Lᵖ and L^q norms of the weight function.