Size-biased branching population measures and the multi-type $x\log x$ condition

Abstract

We investigate the $x\log x$ condition for a general (Crump--Mode--Jagers) multi-type branching process with a general type space by constructing a size-biased population measure that relates to the ordinary population measure via an intrinsic martingale $W_t$. Sufficiency of the $x\log x$ condition for a non-degenerate limit of $W_t$ is proved and conditions for necessity are investigated.

Published in at http://dx.doi.org/10.3150/09-BEJ211 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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