For n equidistant observations of a Lévy process at time distance Δn we consider the problem of testing hypotheses on the volatility, the jump measure and its Blumenthal–Getoor index in a non- or semiparametric manner. Asymptotically as n→∞ we allow for both, the high-frequency regime...

We introduce power variation constructed from powers of the second-order differences of a discretely observed pure-jump semimartingale processes. We derive the asymptotic behavior of the statistic in the setting of high-frequency observations of the underlying process with a fixed time span....