= Jump activity of a Lévy process
The <jump activity of a Lévy process> describes whether it has finitely or infinitely many jumps in a bounded time <interval>. Its <Lévy measure> $\nu$ decides this: finite activity corresponds to $\nu(\mathbb R^d\setminus\{0\})<\infty$, in which case the jump component is a <Compound Poisson process>. Infinite mass gives infinitely many small jumps in every positive-length time <interval>, <almost surely>. For every $\varepsilon>0$, however, jumps larger than $\varepsilon$ still have a finite rate.
Activity and <total variation> are different. With zero Brownian component, <sample paths> have <finite variation> on compact time <intervals> exactly when
$$
\int_{\|x\|\leq1}\|x\|\,\nu(dx)<\infty.
$$
For example, $\nu(dx)=c x^{-1-\alpha}\,dx$ on $(0,\infty)$, with $c>0$ and $0<\alpha<1$, has infinite total mass but a finite integral of $x$ near zero. It therefore permits infinite jump activity with <finite variation>. Conversely, <Brownian motion> has no jumps but infinite <total variation> on every positive-length time <interval>, <almost surely>.<Topics in Probability: Lévy Processes by Davar Khoshnevisan>[Lecture 7, the compound Poisson part and Theorem 5 on path variation]{ref}
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