  With an aim to analyse the performance of Markov chain Monte Carlo (MCMC)
methods, in our recent work we derive a large deviation principle (LDP) for the
empirical measures of Metropolis-Hastings (MH) chains on a continuous state
space. One of the (sufficient) assumptions for the LDP involves the existence
of a particular type of Lyapunov function, and it was left as an open question
whether or not such a function exists for specific choices of MH samplers. In
this paper we analyse the properties of such Lyapunov functions and investigate
their existence for some of the most popular choices of MCMC samplers built on
MH dynamics: Independent Metropolis Hastings, Random Walk Metropolis, and the
Metropolis-adjusted Langevin algorithm. We establish under what conditions such
a Lyapunov function exists, and from this obtain LDPs for some instances of the
MCMC algorithms under consideration. To the best of our knowledge, these are
the first large deviation results for empirical measures associated with
Metropolis-Hastings chains for specific choices of proposal and target
distributions.
