A Central Limit Theorem for Convex Chains in the Square

Abstract

Points P 1 ,... ,P n in the unit square define a convex n -chain if they are below y=x and, together with P 0 =(0,0) and P n+1 =(1,1) , they are in convex position. Under uniform probability, we prove an almost sure limit theorem for these chains that uses only probabilistic arguments, and which strengthens similar limit shape statements established by other authors. An interesting feature is that the limit shape is a direct consequence of the method. 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.