Higher Order Limit Theorems

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


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.


Limit Theorem Central Limit Theorem Degenerate Case Functional Versus Degeneracy Condition 
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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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