Acedemic advisor of the HDI Lab, prof. Eric Moulines, gave a mini-course "Introduction to Markov chains and Markov processes"

Mini-course content and recommended material from the book Markov Chains by Douc, R., Moulines, E., Priouret, P., Soulier, P.

• Chapter 1 : basic definitions and constructions, invariant distribution, reversibility
• Chapter 2 : examples of Markov chains

Suggested reading: beginning of Chapter 3 [Strong Markov properties], Chapter 4 [a glimpse at potential theory, Comparison theorem, Dynkin formula]

• Chapter 18: Uniform and V-uniform ergodicity by operator methods

Suggested reading: Chapter 19: coupling for irreducible kernel [the 4 first sections] 
If you want to understand more what is a small set : Chapter 9 [first section]