Drift burst test statistic in a pure jump semimartingale model

We complete the investigation on the asymptotic behavior of the drift burst test statistic devised in Christensen, Oomen and Ren\`o (2020). They analysed it for an Ito semimartingale containing a Brownian component and finite variation jumps. We also account for infinite variation jumps.We show that when there are no bursts in drift neither in volatility, explosion of the statistic only can occur in the absence of the Brownian part and when the jumps have finite variation. In that case the explosion is due to the compensator of the small jumps.We also find that the statistic could be adopted for a variety of tests useful for investigating the nature of the process, given discrete observations