On largest offspring in a critical branching process with finite variance.

• Autores: Jean Bertoin
• Localización: Journal of Applied Probability, ISSN-e 0021-9002, Vol. 50, Nº. 3, 2013, págs. 791-800
• Idioma: inglés
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• Resumen

  • Continuing the work in Bertoin (2011) we study the distribution of the maximal number Xk* of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail ... with index -a for a > 2 (and, hence, finite variance). We show that Xk* suitably normalized converges in distribution to a Fréchet law with shape parameter a/2; this contrasts sharply with the case 1 < a < 2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.