Geometric convergence of the Metropolis-Hastings simulation algorithm

Necessary and sufficient conditions for geometric convergence of the Metropolis-Hasting simulation algorithm with a general generation function are established. If these conditions are violated, then the algorithm does not in general converge. An explicit expression for the convergence rate is found. The convergence rate depends heavily on the size of the domain where the generation function and the limiting function is this domain and the number of jumps necessary to jump between two arbitrary states. The result in the paper also give a qualitative understanding of the convergence rate.