ABSTRACT
This chapter describes the theory that provides the foundations of statistical modeling for some of the most important random sequences such as random walks and Markov chains. It is amazing to think how many real-life processes exhibit a Markovian property, that is, the future state of the random object is independent of past states given its present state. Martingales play an important role in mathematics and statistics, and can be used as a tool to define more complicated random objects, including stochastic integrals. The chapter provides the definition of a filtration and an adapted sequence of random variables to the filtration. Studying the asymptotic behavior of martingales is due to Doob and his convergence theorem, which provides a great tool to assess the existence of a limit distribution for a random sequence. The proof of the limit theorem requires the concept of upcrossings by the random sequence and the corresponding upcrossings theorem.
