Stein kernels and moment maps

Abstract

We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge-Ampère equation. As a consequence, we show how regularity bounds on these maps control the rate of convergence in the classical central limit theorem, and derive new rates in Kantorovitch-Wasserstein distance in the log-concave situation, with explicit polynomial dependence on the dimension.

v2: improved dependence on the dimension in the quantitative CLT. v3: corrected a wrong sentence, which does not affect the results or the proofs

References in corpus (2)

• Kahler geometry on toric manifolds, and some other manifolds with large symmetry
• Moment measures and stability for Gaussian inequalities

Cited by in corpus (9)

• Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles
• Stein operators, kernels and discrepancies for multivariate continuous distributions
• High-dimensional Central Limit Theorems by Stein's Method
• Regularity of solutions of the Stein equation and rates in the multivariate central limit theorem
• A CLT in Stein's distance for generalized Wishart matrices and higher order tensors
• The Brownian transport map
• The deficit in the Gaussian log-Sobolev inequality and inverse Santalo inequalities
• Malliavin-Stein Method: a Survey of Recent Developments
• On infinite covariance expansions