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The necklace process. (English) Zbl 1213.60027
Summary: Start with a necklace consisting of one white bead and one black bead, and add new beads one at a time by inserting each new bead between a randomly chosen adjacent pair of old beads, with the proviso that the new bead will be white if and only if both beads of the adjacent pair are black. Let \(W_n\) denote the number of white beads when the total number of beads is \(n\). We show that \(\mathrm{E}W_n = n/3\) and, with \(c^{2} = \frac{2}{45}\), that \((W_n - n/3) / c\sqrt n\) is asymptotically standard normal. We find that, for all \(r \geq 1\) and \(n > 2r\), the \(r\)-th cumulant of the distribution of \(W_n\) is of the form \(nh_r\). We find the expected numbers of gaps of given length between white beads, and examine the asymptotics of the longest gaps.

MSC:
60C05 Combinatorial probability
62M05 Markov processes: estimation; hidden Markov models
60F05 Central limit and other weak theorems
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