Cyclic behavior of maxima in a hierarchical summation scheme

TL;DR

This paper investigates the asymptotic distribution of the maximum path sum in a binary tree with Bernoulli variables, revealing a novel cyclic pattern in the distributional limits as the tree depth increases.

Contribution

It introduces a new perspective on the limiting behavior of maxima in hierarchical sums, showing convergence to a helix in the space of distributions.

Findings

Distribution of M_n approaches a helix as n grows

Each distribution shift of M_n converges to an element on the helix

Reveals cyclic behavior in the maxima of hierarchical sums

Abstract

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.