April 2023 A probability approximation framework: Markov process approach
Peng Chen, Qi-Man Shao, Lihu Xu
Author Affiliations +
Ann. Appl. Probab. 33(2): 1619-1659 (April 2023). DOI: 10.1214/22-AAP1853


We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated Itô’s formula and Markov operator. As applications, we study the error bounds of the following three approximations: approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by multiplicative Brownian motion, Euler–Maruyama (EM) discretization for multi-dimensional Ornstein–Uhlenbeck stable process and multivariate normal approximation. All these error bounds are in Wasserstein-1 distance.

Funding Statement

Shao Q.M. is partially supported by National Nature Science Foundation of China NSFC 12031005, Shenzhen Outstanding Talents Training Fund.
Xu L. is partially supported by NSFC No. 12071499, Macao S.A.R grant FDCT 0090/2019/A2 and University of Macau grants (MYRG2018-00133-FST, MYRG2020-00039-FST).


The authors would like to gratefully thank Jim Dai for very helpful discussions on probability approximations. We are grateful to the referees whose constructive comments and suggestions have helped to greatly improve the quality of this paper.

Xu L. is the corresponding author.


Download Citation

Peng Chen. Qi-Man Shao. Lihu Xu. "A probability approximation framework: Markov process approach." Ann. Appl. Probab. 33 (2) 1619 - 1659, April 2023. https://doi.org/10.1214/22-AAP1853


Received: 1 August 2021; Revised: 1 April 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 07692299
MathSciNet: MR4564436
Digital Object Identifier: 10.1214/22-AAP1853

Primary: 60F05 , 60H07
Secondary: 60J20

Keywords: Euler–Maruyama (EM) discretization , Itô’s formula , Markov process , Normal approximation , online stochastic gradient descent , probability approximation , Stable process , Stochastic differential equation , Wasserstein-1 distance

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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