Cyclic behavior of maxima for sums of independent variables

TL;DR

This paper explores the cyclic behavior of maxima in sums of independent variables, revealing that the distribution of the maximum approaches a helix in the distribution space as the number of variables increases.

Contribution

It demonstrates the presence of cyclic behavior in maxima for sums of independent variables, extending previous hierarchical findings to conventional summation schemes.

Findings

Distribution of maxima approaches a helix in distribution space

Cyclic behavior observed in maxima of sums of independent variables

Phenomenon similar to hierarchical schemes also appears in conventional sums

Abstract

In a recent author's work the cyclic behavior of maxima in a hierarchical summation scheme was discovered. In the present note we show how the same phenomenon appears in the scheme of conventional summation: the distribution of maximum of $2^n$ independent copies of a sum of $n$ i.i.d. random variables approaches, as $n$ grows, some helix in the space of distributions.