Random Geometric Series
E. Ben-Naim and P.L. Krapivsky
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 \hbox{$0\leq
p\leq n-1$} is studied. At large $n$, the moments grow
algebraically, $\langle x_n^s\rangle\sim n^{\beta(s)}$ with
$\beta(s)=2^s-1$, while the typical behavior is $x_n\sim n^{\ln
2}$. The probability distribution is obtained explicitly in terms of
the Stirling numbers of the first kind and it approaches a log-normal
distribution asymptotically.
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