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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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