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The diminishing segment process. (English) Zbl 1230.60020
Summary: Let $\Xi _{0}=[-1,1]$, and define the segments $\Xi _{n}$ recursively in the following manner: for every $n=0,1,\ldots $, let $\Xi _{n+1}=\Xi _{n}\cap [a_{n+1} - 1,a_{n+1}+1]$, 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 $n(\rho _{n} - 1/2)$ converges in distribution to an exponential law, and we show that the centre of the limiting unit interval has arcsine distribution.

60F05Central limit and other weak theorems
60J05Discrete-time Markov processes on general state spaces
Full Text: DOI
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