On sequences of convex records in the plane
Claude Godr\`eche, Jean-Marc Luck

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 -dimensional data points, the -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 of convex records up to time to the mean number of vertices in the convex hull of the first 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 and grow…
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Taxonomy
TopicsFunctional Equations Stability Results · Point processes and geometric inequalities · Limits and Structures in Graph Theory
