Higher Order Limit Theorems

Abstract

In some cases, the previous Laws of Large Numbers and/or Central Limit Theorems are degenerate, in the sense that the limiting process is identically 0. In these cases, there is a need for a different normalization, which hopefully leads to a non-degenerate limit. Sect. 15.1 presents a few situations of this type.

A general theory for these cases is currently out of reach, but in Sect. 15.2 we consider a specific degenerate case, which might serve as an example for more complicated cases. Namely, we consider unnormalized functionals depending on k successive increments, in a case where the limit in the Central Limit Theorem of Chap.  11 with normalizing factor \(\sqrt {\varDelta _{n}}\) vanishes identically. We then give two different Central Limit Theorems with normalizing factor Δ _n , for which the limits are non-degenerate, in two slightly different cases.

Section 15.3 is devoted to analyzing whether or not a two-dimensional process X is such that the two components have jumps at the same (random) times.