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Sticky central limit theorems on open books. (English) Zbl 1293.60006

The open book is a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes. The glued hyperplane is called the spine. The authors define the barycenter of a set of points from an open book as the minimizer of the sum of squared distances to the points.
The law of large numbers (LLN) and the central limit theorem (CLT) are considered for the barycenter of i.i.d. random points from an open book. Such a barycenter can be sticky in the sense that it eventually almost surely lies on the spine. Then the limit Gaussian distribution in CLT also is supported by the spine. LLN and CLT for the non-sticky case (when the limit lies not on the spine) and for the partly sticky case (when it lies on the spine but is not sticky) are also considered.

MSC:

60B05 Probability measures on topological spaces
60B99 Probability theory on algebraic and topological structures
60F05 Central limit and other weak theorems

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