On the Stability and Ergodicity of an Adaptive Scaling Metropolis Algorithm

This paper considers the stability and ergodicity of an adaptive random walk Metropolis algorithm. The algorithm adjusts the scale of the symmetric proposal distribution continuously, based on the observed acceptance probability. 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.