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

A central limit theorem for convex chains in the square

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


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.
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