Point processes subordinated to compound Poisson processes

Point processes $N^f (t)=N\bigl (H^f(t)\bigr )$, $t>0$, are studied in the paper where $N(t)$ is a Poisson process and $H^f(t)$ is a subordinator with the Berns̆tein function $f(\lambda )$. We present the probability distribution and moments of the first and second order of processes $N^f(t)$ for the case where $H^f(t)$ is a compound Poisson process with gamma distributed jumps. We also consider these processes with double and iterated time change.