Universality of the Langevin diffusion as scaling limit of a family of Metropolis-Hastings processes I: fixed dimension
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Given a target distribution μ on a general state space X and a proposal Markov jump process with generator Q, the purpose of this paper is to investigate two universal properties enjoyed by two types of Metropolis-Hastings (MH) processes with generators M_1(Q,μ) and M_2(Q,μ) respectively. First, we motivate our study of M_2 by offering a geometric interpretation of M_1, M_2 and their convex combinations as L^1 minimizers between Q and the set of μ-reversible generators of Markov jump processes. Second, specializing into the case of X = R^d along with a Gaussian proposal with vanishing variance and Gibbs target distribution, we prove that, upon appropriate scaling in time, the family of Markov jump processes corresponding to M_1, M_2 or their convex combinations all converge weakly to an universal Langevin diffusion. While M_1 and M_2 are seemingly different stochastic dynamics, it is perhaps surprising that they share these two universal properties. These two results are known for M_1 in Billera and Diaconis (2001) and Gelfand and Mitter (1991), and the counterpart results for M_2 and their convex combinations are new.
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