  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.
