The Central Limit Theorem for Functions of a Finite Number of Increments

Abstract

In this chapter we give the Central Limit Theorems associated with the Laws of Large Numbers of Chap. 8, when the number k of increments in the test function is fixed.

For unnormalized functionals, studied in Sect. 11.1, this is a rather straightforward extension of the Central Limit Theorems given in Chap. 5.

In Sect. 11.2, normalized functionals are considered. In this case, the situation is much more complicated than in Chap. 5, because two successive summands in the definition of the functional involve k−1 “common” increments of the process X. Joint Central Limit Theorems for the two types of functionals are presented in Sect. 11.3.

Finally, in Sect. 11.4 we present some statistical applications, both for the estimation of the volatility and for the detection of jumps of the process X. Using functions of several increments (and in particular multipower variations) allows one to estimate the integrated volatility even when the process X has jumps.