Stochastic stable population theory with continuous time. I

This paper contains a systematic presentation of time-continuous stable population theory in modern probabilistic dress. The life-time births of an individual are represented by an inhomogeneous Poisson process stopped at death, and an aggregate of such processes on the individual level constitutes the population process. Forward and backward renewal relations are established for the first moments of the main functionals of the process and for their densities. Their asymptotic convergence to a stable form is studied, and the stable age distribution is given some attention. It is a distinguishing feature of the present paper that rigorous proofs are given for results usually set up by intuitive reasoning only.