The Critical Galton-Watson Process Without Further Power Moments

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, whi...

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Veröffentlicht in:Journal of applied probability 2007-09, Vol.44 (3), p.753-769
Hauptverfasser: Nagaev, S. V., Wachtel, V.
Format: Artikel
Sprache:eng
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Zusammenfassung:In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.
ISSN:0021-9002
1475-6072
DOI:10.1239/jap/1189717543