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We provide asymptotic results and develop high frequency statisticalprocedures for time-changed L\'evy processes sampled at random instants. Thesampling times are given by first hitting times of symmetric barriers whosedistance with respect to the starting point is equal to $\varepsilon$. Thissetting can be seen as a first step towards a model for tick-by-tick financialdata allowing for large jumps. For a wide class of L\'evy processes, weintroduce a renormalization depending on $\varepsilon$, under which the L\'evyprocess converges in law to an $\alpha$-stable process as $\varepsilon$ goes to$0$. The convergence is extended to moments of hitting times and overshoots. Inparticular, these results allow us to construct consistent estimators of thetime change and of the Blumenthal-Getoor index of the underlying L\'evyprocess. Convergence rates and a central limit theorem are established underadditional assumptions.

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