The Central Limit Theorem for Functions of an Increasing Number of Increments

Abstract

Here we study the same problem as in the previous chapter, except that now the functionals depend on an increasing number k _n of increments, with kj _n →∞ and k _n Δ _n →0.

In this setting, the Central Limit Theorems are considerably more difficult to prove, and the rate of convergence becomes \(\sqrt{k_{n} \varDelta _{n}}\) instead of \(\sqrt {\varDelta _{n}}\). Unnormalized and normalized functionals are studied in Sects. 12.1 and 12.2, respectively.

No specific application is given in this chapter, but it is a necessary step for studying semimartingales contaminated by an observation noise, and we treat this in Chap.  16.