We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated It\^{o}'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.
@article{arxiv.2011.10985,
title = {A probability approximation framework: Markov process approach},
author = {Peng Chen and Qi-Man Shao and Lihu Xu},
journal= {arXiv preprint arXiv:2011.10985},
year = {2022}
}