LangevinType Models I: Diffusions with Given Stationary Distributions and their Discretizations*
 O. Stramer,
 R. L. Tweedie
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We describe algorithms for estimating a given measure π known up to a constant of proportionality, based on a large class of diffusions (extending the Langevin model) for which π is invariant. We show that under weak conditions one can choose from this class in such a way that the diffusions converge at exponential rate to π, and one can even ensure that convergence is independent of the starting point of the algorithm. When convergence is less than exponential we show that it is often polynomial at verifiable rates. We then consider methods of discretizing the diffusion in time, and find methods which inherit the convergence rates of the continuous time process. These contrast with the behavior of the naive or Euler discretization, which can behave badly even in simple cases. Our results are described in detail in one dimension only, although extensions to higher dimensions are also briefly described.
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 Title
 LangevinType Models I: Diffusions with Given Stationary Distributions and their Discretizations*
 Journal

Methodology And Computing In Applied Probability
Volume 1, Issue 3 , pp 283306
 Cover Date
 19991001
 DOI
 10.1023/A:1010086427957
 Print ISSN
 13875841
 Online ISSN
 15737713
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Markov chain Monte Carlo
 diffusions
 Langevin models
 posterior distributions
 irreducible Markov processes
 exponential ergodicity
 uniform ergodicity
 Euler schemes
 Industry Sectors
 Authors

 O. Stramer ^{(1)}
 R. L. Tweedie ^{(2)}
 Author Affiliations

 1. Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA, 52242, USA
 2. Division of Biostatistics, University of Minnesota, Minneapolis, MN, 55455, USA