I have been reading the paper by Moustakides on the extension of Wald's lemma to Markov processes. The paper uses notation that is followed in Meyn and Tweedie's book on Markov processes and stochastic stability, and ventures into the Foster Lyapunov drift criteria, something that I find a bit difficult to follow.
Can anyone help me with good references that build up material towards these drift conditions in a manner that can be picked up by a grad student having background in measure theoretic probability?
The paper by Jones and Hobert explains "drift" and "minorization" in the context of Markov chains, from the perspective of Markov chain Monte Carlo. Although their main reference is Meyn and Tweedie, they take their time to explain the details, and provide a wonderful set of examples.
The drift condition they use is slightly different from the usual Foster-Lyapunov drift, but in this Annals of Statistics paper Jones and Hobert (2004), they explain the relationship between the drift conditions, and also study the drift and minorization for a specific example.
In general, early papers by Jones and a lot of papers by Hobert use drift and minorization, all from the perspective of MCMC. You can also find a lot of work on this by Jeff Rosenthal, specially before 2000s. His perspective is often not limited to MCMC.