Convergence and perturbation theory for an infinite-dimensional Metropolis-Hastings algorithm with self-decomposable priors

We study a Metropolis-Hastings algorithm for target measures that are absolutely continuous with respect to a large class of prior measures on Banach spaces. The algorithm is shown to have a spectral gap in a Wasserstein-like semimetric weighted by a Lyapunov function. A number of error bounds are given for computationally tractable approximations of the algorithm including bounds on the closeness of Ces\'{a}ro averages and other pathwise quantities. Several applications illustrate the breadth of problems to which the results apply such as discretization by Galerkin-type projections and approximate simulation of the proposal via perturbation theory.