ABSTRACT
This chapter gives a rather informal introduction to statistics of stochastic processes. It starts with explaining a “key point” in statistics. The “key point” is closely related to the essence of martingales, and it will be called the “core of statistics” in this monograph. Although this terminology is the one invented for our discussion, there would be no doubt about the importance and usefulness of the point itself.
The main purpose of this chapter is to give an overview of the martingale theory towards applications to statistics. Although the rigorous descriptions will start formally from the next chapter, this chapter already includes some explanations of the importance of stochastic integrals and martingale central limit theorems in statistics. The outline of the proofs of asymptotic normality of the maximum likelihood estimators in stochastic process models will be presented, with the emphasis on the role of martingales.
The chapter finishes with exhibiting some concrete examples of counting and diffusion process models.
