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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