  This article develops general conditions for weak convergence of adaptive
Markov chain Monte Carlo processes and is shown to imply a weak law of large
numbers for bounded Lipschitz continuous functions. This allows an estimation
theory for adaptive Markov chain Monte Carlo where previously developed theory
in total variation may fail or be difficult to establish. Extensions of weak
convergence to general Wasserstein distances are established along with a weak
law of large numbers for possibly unbounded Lipschitz functions. Applications
are applied to auto-regressive processes in various settings, unadjusted
Langevin processes, and adaptive Metropolis-Hastings.
