Annals of Statistics

Identifying the successive Blumenthal–Getoor indices of a discretely observed process

Yacine Aït-Sahalia and Jean Jacod

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This paper studies the identification of the Lévy jump measure of a discretely-sampled semimartingale. We define successive Blumenthal–Getoor indices of jump activity, and show that the leading index can always be identified, but that higher order indices are only identifiable if they are sufficiently close to the previous one, even if the path is fully observed. This result establishes a clear boundary on which aspects of the jump measure can be identified on the basis of discrete observations, and which cannot. We then propose an estimation procedure for the identifiable indices and compare the rates of convergence of these estimators with the optimal rates in a special parametric case, which we can compute explicitly.

Article information

Ann. Statist., Volume 40, Number 3 (2012), 1430-1464.

First available in Project Euclid: 5 September 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Semimartingale Brownian motion jumps finite activity infinite activity discrete sampling high frequency


Aït-Sahalia, Yacine; Jacod, Jean. Identifying the successive Blumenthal–Getoor indices of a discretely observed process. Ann. Statist. 40 (2012), no. 3, 1430--1464. doi:10.1214/12-AOS976.

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Supplemental materials

  • Supplementary material: Supplement to “Identifying the successive Blumenthal–Getoor indices of a discretely observed process”. This supplement contains the proof of Theorem 4.