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Cyclic behavior of maxima in a hierarchical summation scheme

Let i.i.d. symmetric Bernoulli random variables be associated to the edges of a binary tree having n levels. To any leaf of the tree, we associate the sum of variables along the path connecting the leaf with the tree root. Let M_n denote the maximum of all such sums. We prove that, as n grows, the distributions of M_n approach some helix in the space of distributions. Each element of this helix is an accumulation point for the shifts of distributions of M_n.