My goal is to get a firm theoretical foundation for the Metropolis-Hastings algorithm and the Gibbs sampler, but none of my classes or books I have found have covered this.
I found this paper titled ON THE CONVERGENCE OF THE MARKOV CHAIN SIMULATION METHOD by Athreya, Doss and Sethuraman which proves some things about limiting distributions of Markov chains with general state space, but it seemed pretty technical. Since I don't have a lot of free time to try and work through it, I was thinking maybe there is a more straightforward proof for a chain the context of the Gibbs sampler where the state space is $\mathbb{R}^n$ and the random variables have a density with respect to some product of Lebesgue measure and counting measure
Is anyone familiar with a more straightforward proof In this context?