The Central Limit Theorem for Functions of an Increasing Number of Increments

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


Here we study the same problem as in the previous chapter, except that now the functionals depend on an increasing number k n of increments, with kj n →∞ and k n Δ n →0.

In this setting, the Central Limit Theorems are considerably more difficult to prove, and the rate of convergence becomes \(\sqrt{k_{n} \varDelta _{n}}\) instead of \(\sqrt {\varDelta _{n}}\). Unnormalized and normalized functionals are studied in Sects. 12.1 and 12.2, respectively.

No specific application is given in this chapter, but it is a necessary step for studying semimartingales contaminated by an observation noise, and we treat this in Chap.  16.


Central Limit Theorem Homogeneous Function Polynomial Growth Auxiliary Space Stable Convergence 
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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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