Asymptotic results and statistical procedures for time-changed L\'evy processes sampled at hitting times

We provide asymptotic results and develop high frequency statistical procedures for time-changed L\'evy processes sampled at random instants. The sampling times are given by first hitting times of symmetric barriers whose distance with respect to the starting point is equal to $\varepsilon$. This setting can be seen as a first step towards a model for tick-by-tick financial data allowing for large jumps. For a wide class of L\'evy processes, we introduce a renormalization depending on $\varepsilon$, under which the L\'evy process converges in law to an $\alpha$-stable process as $\varepsilon$ goes to $0$. The convergence is extended to moments of hitting times and overshoots. In particular, these results allow us to construct consistent estimators of the time change and of the Blumenthal-Getoor index of the underlying L\'evy process. Convergence rates and a central limit theorem are established under additional assumptions.