ABSTRACT
This chapter is devoted to preparing three tools for asymptotic statistics—the martingale central limit theorems, the uniform convergence of random fields, and some techniques concerning discrete sampling of diffusion processes. Many of the theorems presented in this chapter could be generalized more; for example, the first topic, the limit theory for (semi-)martingales, is exposited much in depth by some authoritative books such as Hall and Heyde (1980, Academic Press) and Jacod and Shiryaev (2003, Springer). In contrast, we aim to give some elementary, self-contained proofs to which readers can easily access. The tools prepared here will be applied to concrete problems in statistics in the subsequent chapters repeatedly.
