Limit theorems for random polytopes generated by heavy-tailed distributions

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

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", St. Petersburg Branch

Abstract: The paper is devoted to the study of asymptotic properties of random polytopes that are convex hulls of independent identically distributed random vectors with a regularly varying (heavy-tailed) distribution. We study the convergence of functionals of these random polytopes, including intrinsic volumes, the induced $U$-max statistics and the $f$-vector, to the corresponding functionals of Poisson polytopes. The results obtained extend known facts for specific distributions to a general class of heavy-tailed distributions.

Key words and phrases: random polytopes, convex hull, regularly varying distributions, heavy tails, Poisson point process, intrinsic volumes, valuations, $U$-max statistics, $f$-vector.

UDC: 519.2

Received: 17.11.2025