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Imre Bárány, Günter Rote, William Steiger, and Cun-Hui
Zhang:

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A central limit theorem for convex chains in the square

*Discrete and Computational Geometry* **23** (2000), 35–50. doi:10.1007/PL00009490
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Abstract

We consider the probability that n points drawn uniformly at random from
the unit square form a convex chain together with the two corners (0,0)
and (1,1). Conditioned under this event, these chains converge to a parabolic
limit shape. We even get an almost sure limit theorem, which uses only
probabilistic arguments and which strengthens similar limit shape statements
established by other authors. The main result is an accompanying central
limit theorem for these chains. A weak convergence result implies several
other statements concerning the deviations between random convex chains
and their limit.