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Higher Order Limit Theorems

  • Jean Jacod
  • Philip Protter
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 67)

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.

Keywords

Limit Theorem Central Limit Theorem Degenerate Case Functional Versus Degeneracy Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité Paris VI – Pierre et Marie CurieParisFrance
  2. 2.Department of StatisticsColumbia UniversityNew YorkUSA

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