## The Annals of Probability

### Stein kernels and moment maps

Max Fathi

#### 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 in certain weighted Sobolev spaces 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2172-2185.

Dates
Revised: August 2018
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205706

Digital Object Identifier
doi:10.1214/18-AOP1305

Mathematical Reviews number (MathSciNet)
MR3980918

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 35J96: Elliptic Monge-Ampère equations

#### Citation

Fathi, Max. Stein kernels and moment maps. Ann. Probab. 47 (2019), no. 4, 2172--2185. doi:10.1214/18-AOP1305. https://projecteuclid.org/euclid.aop/1562205706

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