As described by the title, I would like to know if all discontinuous Levy processes (or jump Levy process, such as Poisson Process, Cauchy process, Gamma Process, etc.) are sparse processes?
The sparse process, as far as I know, is defined as $$\lim_{dt\rightarrow 0} \frac{P(X(dt)=k)}{P(X(dt)=1)}= 0$$ for all $k\geq 2$, which means the sample trajectory cannot jump more than twice in a very small time period.
If the above proposition is true, how to prove it?