Random Geometric Series
Abstract
Integer sequences where each element is determined by a previous randomly chosen element are investigated analytically. In particular, the random geometric series xn=2xp with 0<=p<=n-1 is studied. At large n, the moments grow algebraically, <xns> nbeta(s) with beta(s)=2s-1, while the typical behavior is xn nln 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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