ABSTRACT
In this chapter, we prepare three things. The probability theory, including the martingale theory, is developed based on the measure theory. We thus first recall the usages of some important limit operations in the measure theory.
Secondly, an intuitive explanation for the conditional expectation is presented. Since the martingale is defined by using the conditional expectation, it would be very helpful for readers of this monograph to understand what is the conditional expectation at this stage. We also summarise some important properties of the conditional expectation.
The main theme of this monograph is the asymptotic theory in statistics. Thirdly, we introduce the definitions, relationships, and related theorems concerning the almost sure convergence, the convergence in probability and the convergence in law for random sequences that will be the bases for our study on asymptotic statistics.
