The Annals of Probability

Stein kernels and moment maps

Max Fathi

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Stein’s method central limit theorem optimal transport Monge–Ampère equation


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

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