### Abstract

Let M-n denote the size of the largest amongst the first n generations of a simple branching process. It is shown for near critical processes with a finite offspring variance that the law of M-n/n, conditioned on no extinction at or before n, has a non-defective weak limit. The proof uses a conditioned functional limit theorem deriving from the Feller-Lindvall (CB) diffusion limit for branching processes descended from increasing numbers of ancestors. Subsidiary results are given about hitting time laws for CB diffusions and Bessel processes.

Original language | English |
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Pages (from-to) | 740-756 |

Journal | Advances in Applied Probability |

Volume | 30 |

Publication status | Published - 1998 |