The existence of moments of the conditioned limit of the subcritical Crump-Mode-Jagers branching process (Q1064677)

In the subcritical generalised Crump-Mode-Jagers branching process, counted by a random characteristic, it was shown by \textit{P. J. Green} [J. Appl. Probab. 14, 451-463 (1977; Zbl 0374.60108)], under certain regularity conditions, that the generalised ''size'' at time t, conditional on the population still being alive at that time, converges in distribution to a limit with finite mean. This paper considers higher integral moments of the limit distribution, obtaining necessary and sufficient conditions for their existence in terms of moments of the offspring process.