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 $\mean{R_n}$ of convex records up to time $n$ to the mean number $\mean{N_n}$ 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 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 $\mean{R_n}$ of convex records and demonstrate that the ratio $R_n/\mean{R_n}$ keeps fluctuating with a universal limit distribution.