We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function evaluated at the increments of a Lévy process between the successive times for = 0,1,...,. One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/or centering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen test function . As for the associated central limit theorem, one can show some versions of it, but unfortunately in a limited number of cases only: in some other cases a CLT just does not exist.
Mots clés : central limit theorem, quadratic variation, power variation, Lévy processes
@article{PS_2007__11__173_0,
author = {Jacod, Jean},
title = {Asymptotic properties of power variations of {L\'evy} processes},
journal = {ESAIM: Probability and Statistics},
pages = {173--196},
publisher = {EDP-Sciences},
volume = {11},
year = {2007},
doi = {10.1051/ps:2007013},
mrnumber = {2320815},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps:2007013/}
}
TY - JOUR AU - Jacod, Jean TI - Asymptotic properties of power variations of Lévy processes JO - ESAIM: Probability and Statistics PY - 2007 SP - 173 EP - 196 VL - 11 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2007013/ DO - 10.1051/ps:2007013 LA - en ID - PS_2007__11__173_0 ER -
Jacod, Jean. Asymptotic properties of power variations of Lévy processes. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 173-196. doi : 10.1051/ps:2007013. http://archive.numdam.org/articles/10.1051/ps:2007013/
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