Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to εε. For a wide class of Lévy processes, we introduce a renormalization depending on εε, under which the Lévy process converges in law to an αα-stable process as εε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal–Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.