Document ESAIM: P&S, June 2007, Vol. 11, pp. 173-196
DOI: 10.1051/ps:2007013
Asymptotic properties of power variations of Lévy processes
Jean JacodInstitut de mathématiques de Jussieu, 175 rue du Chevaleret 75 013 Paris, France (CNRS - UMR 7586, and Université Pierre et Marie Curie-P6); jj@ccr.jussieu.fr
(Received November 4, 2005. Revised April 4, 2006. Published online 19 June 2007.)
Abstract
We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Lévy process between
the successive times 
for i = 0,1,...,n. 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 f. 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.
Mathematics Subject Classification. 60F17, 60G51
Key words: Central limit theorem, quadratic variation, power variation, Lévy processes.
© EDP Sciences, SMAI 2007