On sequences of convex records in the plane

TL;DR

This paper investigates the statistical properties of convex records in two-dimensional data sequences, establishing identities and analyzing growth patterns for various distributions and random walks.

Contribution

It introduces a new identity linking convex records and convex hull vertices, and provides extensive numerical analysis across different data models.

Findings

Mean number of convex records and hull vertices grow proportionally.

Finite Fano factors indicate stable variance-to-mean ratios.

Universal limit distribution for fluctuations in convex records ratio.

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 P\'olya 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.