Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps happen.
There are two questions for me. First, is $\tau_{i}$ well defined?. Second, the answer of first one is yes, we know $\tau_{i}$ are i.i.d. Then how to find the distribution of $\tau_{i}$? Can we prove the expectation of $\tau_{i}$ is finite or even its variance is finite according to the information of Levy measure $\nu$?
I can do this when the Levy process $X_{t}$ is Poisson process or negative binomial process. I have difficulty when the probability density function of $\nu$ is continuous. For example, the case that $X_{t}$ is a Gamma process.
Any references would be very appreciated.