Assume that we have a temporal point process on the interval [0, 1], where the number of events $N$ is distributed according to some PMF $p(N = n)$. Conditioned on $N$, the arrivals times of the events $t_i$ (for $i = 1, \dots, N$) are sampled independently from a distribution with a PDF $f(t)$, supported on the interval $[0, 1]$.
How can I obtain the conditional intensity $\lambda^*(t)$ of this point process?
I know that in the special case of $p(N = n) =\textrm{Poisson}(n; \mu)$ and $f(t) = \textrm{Uniform}(t; [0, 1])$ we obtain the Poisson process with conditional intensity $\lambda^*(t) = \mu$. Similarly, in case $p(N = n) =\textrm{Poisson}(n; \mu)$ and $f(t)$ is an arbitrary density on $[0, 1]$, we obtain the inhomogeneous Poisson process with conditional intensity $\lambda^*(t) = \mu f(t)$ (see, e.g., Theorem 2 in this paper). I wonder if this result can be generalized to an arbitrary PMF $p(N = n)$.
I'm also not sure how such generalization of the Poisson process is called in the literature, and I will appreciate any pointers in this direction. So far, I found only one paper (first paragraph of Section 4.3) that refers to this generalized Poisson process as "cluster models" and cites "An Introduction to the Theory of Point Processes, Vol I" by Daley & Vere-Jones. However, I was unable to find any references to "cluster models" in the book.