Markov Chain Monte Carlo on Finite State Spaces

We elaborate the idea behind Markov chain Monte Carlo (MCMC) methods in a mathematically coherent, yet simple and understandable way. To this end, we give elementary proofs for the Perron-Frobenius theorem and a necessary convergence theorem for Markov chains. Subsequently, we briefly discuss the well-known Gibbs sampler and the Metropolis-Hastings algorithm. Only very basic knowledge about matrices, convergence of real sequences and stochastics is required.