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The diminishing segment process. (English) Zbl 1230.60020
Summary: Let Ξ 0 =[-1,1], and define the segments Ξ n recursively in the following manner: for every n=0,1,..., let Ξ n+1 =Ξ n [a n+1 -1,a n+1 +1], where the point a n+1 is chosen randomly on the segment Ξ n with uniform distribution. For the radius ρ n of Ξ n , we prove that n(ρ n -1/2) converges in distribution to an exponential law, and we show that the centre of the limiting unit interval has arcsine distribution.
MSC:
60F05Central limit and other weak theorems
60J05Discrete-time Markov processes on general state spaces
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[5]Tóth, B., Private communication, 2010.