  This paper considers population processes in which general, not necessarily
Markovian, multivariate Hawkes processes dictate the stochastic arrivals. We
establish results to determine the corresponding time-dependent joint
probability distribution, allowing for general intensity decay functions,
general intensity jumps, and general sojourn times. We obtain an exact, full
characterization of the time-dependent joint transform of the multivariate
population process and its underlying intensity process in terms of a
fixed-point representation and corresponding convergence results. We also
derive the asymptotic tail behavior of the population process and its
underlying intensity process in the setting of heavy-tailed intensity jumps. By
exploiting the results we establish, arbitrary joint spatial-temporal moments
and other distributional properties can now be readily evaluated using standard
transform differentiation and inversion techniques, and we illustrate this in a
few examples.
