A PROBABILITY APPROXIMATION FRAMEWORK: MARKOV PROCESS APPROACH

We view the classical Lindeberg principle in a Markov process setting to establish a probability approximation framework by the associated Ito's for-mula 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 nor-mal approximation. All these error bounds are in Wasserstein-1 distance.