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 nth 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 of convex records up to time n to the mean number 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 Nn and Rn grow proportionally to each other, resulting in the finite limit Fano factors FN and FR. 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 of convex records and demonstrate that the ratio keeps fluctuating with a universal limit distribution.