Open Access
June 2012 Identifying the successive Blumenthal–Getoor indices of a discretely observed process
Yacine Aït-Sahalia, Jean Jacod
Ann. Statist. 40(3): 1430-1464 (June 2012). DOI: 10.1214/12-AOS976


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.


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Yacine Aït-Sahalia. Jean Jacod. "Identifying the successive Blumenthal–Getoor indices of a discretely observed process." Ann. Statist. 40 (3) 1430 - 1464, June 2012.


Published: June 2012
First available in Project Euclid: 5 September 2012

zbMATH: 1297.62051
MathSciNet: MR3015031
Digital Object Identifier: 10.1214/12-AOS976

Primary: 62F12 , 62M05
Secondary: 60H10 , 60J60

Keywords: Brownian motion , discrete sampling , finite activity , high frequency , infinite activity , jumps , Semimartingale

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 3 • June 2012
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