The Central Limit Theorem for Truncated Functionals

Abstract

In this chapter we prove the Central Limit Theorems associated with the Laws of Large Numbers of Chap. 9, about truncated functionals.

Section 13.1 is devoted to unnormalized functionals, truncated upward, whereas Sect. 13.2 is about normalized functionals, truncated downward, and we include functionals depending on k successive increments. In both cases, the proofs are straightforward extensions of those in Chaps. 5 and 11.

In Sect. 13.3 one studies the rate of convergence of the “local approximations” of the volatility, introduced in Chap. 9. This part necessitates some novel techniques, and leads to some new and a priori surprising results: for example the normalized error process (the difference between the estimation of σ _t and the process σ _t itself) is asymptotically a white noise.

Applications to the estimation of the integrated volatility are given in Sect. 13.4; in particular a thorough comparison between the methods based on multipower variations and those based on downward truncated functionals is presented.