Geometric Convergence of the Metropolis Hastings Simulation Algorithm Lars Holden Norwegian Computing Center and University of Oslo

Necessary and su cient conditions for geometric con vergence in the relative supremum norm of the Metropolis Hastings simulation algorithm with a general generating function are estab lished An explicit expression for the convergence rate is given Introduction This paper discusses the convergence rate for the Metropolis Hastings simulation algorithm proposed in Hastings The Metropolis Hastings simulation algorithm is used for sampling from a distribution f x There is currently a lot of interest in MCMC both theoretically and in a large number of applications see Geyer The challenge in Metropolis Hastings is to nd a good generating function The explicit formula for the convergence rate given in this paper may be used to compare di erent generating functions Meyn Tweedie prove that the Doeblin condition is equivalent to uniform ergodic i e uniform convergence in total Date April Mathematics Subject Classi cation J