Cyclic behavior of the maximum of sums of independent random variables

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.