Central Limit Theorems: Technical Tools

Abstract

The aim of this chapter is to establish the theoretical basis for the Central Limit Theorems associated with the Laws of Large Numbers of the previous chapter.

The reason for presenting this material in a separate chapter is that Central Limit Theorems have rather long proofs, but for the functionals previously considered, as well as for more general functionals to be seen in the forthcoming chapters, the proofs are always based on the same ideas and techniques; we thus thought more suitable to present these general ideas and technical tools in a specific chapter.

These tools will be in constant use later, hence the results of this chapter are considerably more general than strictly needed for the functionals V ⁿ(f,X) and V′ⁿ(f,X) presented before. However, although this greater generality complicates the notation, it creates only very few supplementary technicalities.

Section 4.1 is devoted to a description of the limiting processes occurring in the various Central Limit Theorems, and which are processes with “conditionally independent increments”. Sections 4.2 and 4.3 provide general criteria for stable convergence in law, when the limit is a continuous process (Sect. 4.3) or a possibly discontinuous one (Sect. 4.3). These results are given for sums of triangular arrays, and the application to discretized Itô semimartingales is presented in Sect. 4.4. This last section also contains a description of the so-called “localization procedure” which allows us to reduce the problem to the case where the semimartingale has nice properties, namely the boundedness of its characteristics.