We prove a diffusion limit theorem in the sense of weak convergence of measure-valued processes for a population age model first studied by Kendall. We show that in the diffusion limit scaling, the population structured in age groups behaves in the same way as the total population size, but with an exponential weight. A particular feature of the limiting process is that in general it is discontinuous at time zero.
"Diffusion Approximation for an Age-Structured Population." Ann. Appl. Probab. 5 (1) 140 - 157, February, 1995. https://doi.org/10.1214/aoap/1177004833