On planar convex records

D. N. Zaporozhets^ab, E. N. Simarova<sup>ca

a</sup> St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
^b Saint Petersburg State University
^c National Research University Higher School of Economics, Moscow

Abstract: Let $X_1, X_2, \ldots$ be a sequence of independent identically distributed random vectors in $\mathbb{R}^2$. A vector $X_n$ is called a convex record if it does not belong to the convex hull of the preceding vectors $\operatorname{conv}(X_1, \ldots, X_{n-1})$. In this paper, we investigate the asymptotic behavior of the mean number of convex records for distributions with exponentially decaying tails.
It is shown that a properly normalized empirical measure of convex records converges weakly in mean to an absolutely continuous limiting measure with an explicitly computed density.

Key words and phrases: convex records, convex hull, spherically symmetric distributions, light tails, Gumbel domain of attraction, random polytopes, vertices of the convex hull, extreme values, empirical measures, weak convergence.

UDC: 519.21, 514.17, 519.218.7, 514.172

Received: 25.12.2025