On sequences of convex records in the plane

Abstract Convex records have an appealing purely geometric definition. In a sequence of d -dimensional data points, the n th point is a convex record if it lies outside the convex hull of all preceding points. We specifically focus on the bivariate (i.e. two-dimensional) setting. For iid (independent and identically distributed) points, we establish an identity relating the mean number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of convex records up to time n to the mean number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of vertices in the convex hull of the first n points. By combining this identity with extensive numerical simulations, we provide a comprehensive overview of the statistics of convex records for various examples of iid data points in the plane: uniform points in the square and in the disk, Gaussian points and points with an isotropic power-law distribution. In all these cases, the mean values and variances of N n and R n grow proportionally to each other, resulting in the finite limit Fano factors F N and F R . We also consider planar random walks, i.e. sequences of points with iid increments. For both the Pearson walk in the continuum and the Pólya walk on a lattice, we characterise the growth of the mean number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of convex records and demonstrate that the ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:math> keeps fluctuating with a universal limit distribution.