A limit theorem for sample maxima and heavy branches in Galton–Watson trees

Let Y, be the maximum of n independent positive random variables with common distribution function F and let S, be their sum. Then Y,S-' converges to zero in probability if and only if fJ xF(dx) is slowly varying. This result implies that in a supercritical Galton-Watson process which does not become extinct, there cannot be a sequence {T,} of particles, each descended from the preceding one, such that the fraction of all particles which are descendants of 7, does not converge to zero as n -oo. Weakly m-adic trees, which behave to some extent like sample Galton-Watson trees, can have such sequences of particles. SAMPLE MAXIMUM; SAMPLE SUM; CONVERGENCE IN PROBABILITY; SLOW VARIATION; SUPERCRITICAL GALTON-WATSON TREE; WEAKLY m -ADIC