ABSTRACT
This chapter summarizes that the main mathematical tools needed to analyse the convergence properties of Markov chains when the time parameter tends to infinity. It presents some more specific advanced probabilistic tools such as coupling, strong stationary times, martingales and martingale limit theorems. The chapter describes the decomposition of a two-state Markov chain transition using its eigenvalue decomposition. It provides with some comments on the diagonalisation techniques. The spectral decomposition theorem applies to any reversible Markov transition on some finite state space. The chapter focuses on the consequences of the spectral decompositions can be found in the seminal Saint Flour summer school lectures of Laurent Saloff-Costes.
