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We obtain results connecting the distributions of the random variables Z 1 and W in the supercritical Galton-Watson process. For example, if a > 1, and converge or diverge together, and regular variation of the tail of one of Z 1 , W with non-integer exponent α > 1 is equivalent to regular variation of the tail of the other.
Bingham et al. (Sun,) studied this question.