1 Probability Theory and Stochastic Processes

In this chapter, we introduce the basic notation and fundamental tools of probability theory and measure theory that will be used throughout the subsequent chapters. The section is organized as follows:

1.

probability space and random variables;

2.

modes of convergence;

3.

the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT);

4.

martingales, Markov chains and Brownian motion.

This section mainly follows Introduction to Probability Theory at Tsinghua University [19], Stochastic Processes and Stochastic Analysis at Peking University [17, 16], and MATH C218A/B at UC Berkeley [4, 20]. I also referred to the books Probability: Theory and Examples [7] and Introduction to Stochastic Processes [11].