High-dimensional normal approximations for sums of Langevin Markov chains

Abstract

Consider the well-known Langevin diffusion on $\mathbb{R}^d$ $$\mathrm{d} X_t = -\nabla U(X_t)\,\mathrm{d} t + \sqrt{2}\mathrm{d} B_t, $$ and its Euler-Maruyama discretization given by $$X_{k+1}=X_k-\eta \nabla U(X_k)+\sqrt{2\eta }\xi_{k+1},$$ where $\eta$ is the step size. Under mild conditions, the Langevin diffusion admits $\pi(\mathrm{d} x)\propto \exp(-U(x))\mathrm{d} x$ as its unique stationary distribution. In this paper, we mainly study the normal approximation of the normalized partial sum $$ W_n = \eta^{1/2} n^{-1/2} \left( \sum_{i=0}^{n-1} X_i- \int_{\mathbb{R}^d} x\,\pi(\mathrm{d} x) \right).$$ To the best of our knowledge, this work provides the first dimension-explicit convergence rates in high-dimensional settings. Our main tool is a novel upper bound for the 1-Wasserstein distance $W_1(W,\gamma)$ via the exchange pair approach, where $W$ is any random vector of interest and $\gamma$ is a $d$-dimensional standard normal random vector.