On the Stability and Ergodicity of an Adaptive Scaling Metropolis Algorithm

The stability and ergodicity properties of an adaptive random walk Metropolis algorithm are considered. The algorithm adjusts the scale of the symmetric proposal distribution continuously based on the observed acceptance probability. Unlike the previously proposed forms of this algorithm, the adapted scaling parameter is not constrained within a predefined compact interval. This makes the algorithm more generally applicable and `automatic,' with two parameters less to be adjusted. A strong law of large numbers is shown to hold for functionals bounded on compact sets and growing at most exponentially as $\|x\|\to\infty$, assuming that the target density is smooth enough and has either compact support or super-exponentially decaying tails.