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The diminishing segment process. (English) Zbl 1230.60020
Summary: Let ${{\Xi }}_{0}=\left[-1,1\right]$, and define the segments ${{\Xi }}_{n}$ recursively in the following manner: for every $n=0,1,...$, let ${{\Xi }}_{n+1}={{\Xi }}_{n}\cap \left[{a}_{n+1}-1,{a}_{n+1}+1\right]$, where the point ${a}_{n+1}$ is chosen randomly on the segment ${{\Xi }}_{n}$ with uniform distribution. For the radius ${\rho }_{n}$ of ${{\Xi }}_{n}$, we prove that $n\left({\rho }_{n}-1/2\right)$ converges in distribution to an exponential law, and we show that the centre of the limiting unit interval has arcsine distribution.
##### MSC:
 60F05 Central limit and other weak theorems 60J05 Discrete-time Markov processes on general state spaces
##### References:
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