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Exponential convergence of Langevin distributions and their discrete approximations. (English) Zbl 0870.60027

Summary: We consider a continuous-time method of approximating a given distribution \(\pi\) using the Langevin diffusion \(d{\mathbf L}_t= d{\mathbf W}_t+{1\over 2}\nabla\log\pi({\mathbf L}_t)dt\). We find conditions under which this diffusion converges exponentially quickly to \(\pi\) or does not: in one dimension, these are essentially that for distributions with exponential tails of the form \(\pi(x)\propto\exp(-\gamma|x|^\beta)\), \(0<\beta<\infty\), exponential convergence occurs if and only if \(\beta\geq 1\). We then consider conditions under which the discrete approximations to the diffusion converge. We first show that even when the diffusion itself converges, naive discretizations need not do so. We then consider a ‘Metropolis-adjusted’ version of the algorithm, and find conditions under which this also converges at an exponential rate: perhaps surprisingly, even the Metropolized version need not converge exponentially fast even if the diffusion does. We briefly discuss a truncated form of the algorithm which, in practice, should avoid the difficulties of the other forms.

MSC:

60F99 Limit theorems in probability theory
60J60 Diffusion processes
65C10 Random number generation in numerical analysis
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