Power variation from second order differences for pure jump semimartingales

Viktor Todorov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


We introduce power variation constructed from powers of the second-order differences of a discretely observed pure-jump semimartingale processes. We derive the asymptotic behavior of the statistic in the setting of high-frequency observations of the underlying process with a fixed time span. Unlike the standard power variation (formed from the first-order differences of the process), the limit of our proposed statistic is determined solely by the jump component of the process regardless of the activity of the latter. We further show that an associated Central Limit Theorem holds for a wider range of activity of the jump process than for the standard power variation. We apply these results for estimation of the jump activity as well as the integrated stochastic scale.

Original languageEnglish (US)
Pages (from-to)2829-2850
Number of pages22
JournalStochastic Processes and their Applications
Issue number7
StatePublished - 2013


  • Blumenthal-Getoor index
  • Jump activity
  • Lévy process
  • Power variation
  • Stable convergence

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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