A universal probability approximation method: Markov process approach

We view the classical Lindeberg principle in a Markov process setting to establish a universal probability approximation framework by It\^{o}'s formula and Markov semigroup. As applications, we consider approximating a family of online stochastic gradient descents (SGDs) by a stochastic differential equation (SDE) driven by additive Brownian motion, and obtain an approximation error with explicit dependence on the dimension which makes it possible to analyse high dimensional models. We also apply our framework to study stable approximation and normal approximation and obtain their optimal convergence rates (up to a logarithmic correction for normal approximation).