On the limit of a supercritical branching process

Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the 'Schroder' and 'B6ttcher' cases; both appear in work of Harris and Dubuc. The Schr6der case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the B6ttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

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