The Central Limit Theorem for Truncated Functionals

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


In this chapter we prove the Central Limit Theorems associated with the Laws of Large Numbers of Chap.  9, about truncated functionals.

Section 13.1 is devoted to unnormalized functionals, truncated upward, whereas Sect. 13.2 is about normalized functionals, truncated downward, and we include functionals depending on k successive increments. In both cases, the proofs are straightforward extensions of those in Chaps.  5 and  11.

In Sect. 13.3 one studies the rate of convergence of the “local approximations” of the volatility, introduced in Chap.  9. This part necessitates some novel techniques, and leads to some new and a priori surprising results: for example the normalized error process (the difference between the estimation of σ t and the process σ t itself) is asymptotically a white noise.

Applications to the estimation of the integrated volatility are given in Sect. 13.4; in particular a thorough comparison between the methods based on multipower variations and those based on downward truncated functionals is presented.


Central Limit Theorem Local Approximation Wiener Process Asymptotic Variance Continuous Part 
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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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