We investigate the x log x condition for a general (Crump–Mode–Jagers) multi-type branching process with arbitrary type space by constructing a size-biased population measure that relates to the ordinary population measure via the intrinsic martingale Wt. Sufficiency of the x log x condition for a non-degenerate limit of Wt is proved and conditions for necessity are investigated.
Bernoulli Society for Mathematical Statistics and Probability
Olofsson, P. (2009). Size-biased branching population measures and the multi-type x log x condition. Bernoulli, 15(4), 1287-1304. doi: 10.3150/09-BEJ211