Annals of Statistics
- Ann. Statist.
- Volume 40, Number 3 (2012), 1430-1464.
Identifying the successive Blumenthal–Getoor indices of a discretely observed process
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.
Ann. Statist., Volume 40, Number 3 (2012), 1430-1464.
First available in Project Euclid: 5 September 2012
Permanent link to this document
Digital Object Identifier
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]
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. https://projecteuclid.org/euclid.aos/1346850061
- Supplementary material: Supplement to “Identifying the successive Blumenthal–Getoor indices of a discretely observed process”. This supplement contains the proof of Theorem 4.