Abstract
By the Berry-Esseen lemma and an important extension of the conditional Borel-Cantelli lemma (Asmussen, Trans Am Math Soc,1977,231:233), we obtain the law of the iterated logarithm of the branching processes in varying environment under the condition that the second moment of the number of the offspring of each individual of each generation is uniformly upper/lower bounded. Further more, the condition is weaker than that of Gao(Gao, UCAS, Thesis 2011).
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