We propose statistical tests to discriminate between the finite and infinite activity of jumps in a semimartingale discretely observed at high frequency. The two statistics allow for a symmetric treatment of the problem: we can either take the null hypothesis to be finite activity, or infinite activity. When implemented on high-frequency stock returns, both tests point toward the presence of infinite-activity jumps in the data.
"Testing whether jumps have finite or infinite activity." Ann. Statist. 39 (3) 1689 - 1719, June 2011. https://doi.org/10.1214/11-AOS873