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 $$X_{k+1}=X_k-η\nabla U(X_k)+\sqrt{2η}ξ_{k+1},$$ where $η$ is the step size. Under mild conditions, the Langevin diffusion admits $π(\mathrm{d} x)\propto \exp(-U(x))\mathrm{d} x$ as its unique stationary distribution. In this paper, we mainly study the normal approximation of the normalized partial sum $$ W_n = η^{1/2} n^{-1/2} \left( \sum_{i=0}^{n-1} X_i- \int_{\mathbb{R}^d} x\,π(\mathrm{d} x) \right).$$ 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 $W_1(W,γ)$ via the exchange pair approach, where $W$ is any random vector of interest and $γ$ is a $d$-dimensional standard normal random vector.

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