Random Geometric Series

TL;DR

This paper analytically investigates a class of integer sequences called random geometric series, revealing their growth behavior, probability distribution, and asymptotic properties, including moments and typical values.

Contribution

It provides an explicit analytical characterization of the distribution and moments of the random geometric series, including their asymptotic behavior and distribution form.

Findings

Moments grow algebraically as n^{2^s-1}

Typical values grow as n^{ln 2}

Distribution approaches a log-normal form asymptotically

Abstract

Integer sequences where each element is determined by a previous randomly chosen element are investigated analytically. In particular, the random geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments grow algebraically, <x_n^s> n^beta(s) with beta(s)=2^s-1, while the typical behavior is x_n n^ln 2. The probability distribution is obtained explicitly in terms of the Stirling numbers of the first kind and it approaches a log-normal distribution asymptotically.