Random Geometric Series
Ben-Naim, E. · Krapivsky, P. L.
Original · EN
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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