Ranges of Extremal Processes and Heavy-Tailed Random Walks in Spaces of Growing Dimension
Bochen Jin, Ilya Molchanov
TL;DR
The paper extends extremal-process and random-walk limit theory to settings where both sample size and ambient dimension grow, using a permutation-invariant counting-measure framework under $\,\ell_p$ geometry. It introduces a tail-measure framework in triangular-array form and identifies limiting objects as a crinkled subordinator tied to a Poisson cluster process, with convergence statements in Hausdorff and Gromov–Hausdorff senses. Three core results (A–C) yield three interrelated limits: the counting-measure images, the limiting subset in $\ell_p$, and the limiting metric space $([0,T],\rho_{\nu,p})$, with examples including iid regularly varying components and long-range dependent time-series vectors. The work provides a robust approach to extremal geometry in growing dimensions and supplies concrete instances that verify the theory.
Abstract
We consider extremal processes and random walks generated by heavy-tailed random vectors taking values in $\mathbb{R}^d$ with the $\ell_p$ distance. We establish limit theorems for their paths in the triangular array setting when both the number of steps $n$ and the dimension $d$ grow to infinity. It is shown that it is possible to map the paths using isometries of $\ell_p$ such that the images converge and to identify the limit as derived from a Poisson cluster process. This also implies the convergence in distribution of the paths interpreted as finite metric spaces on the family of metric spaces equipped with the Gromov-Hausdorff distance. We also show that the images of the paths converge in distribution in the space of counting measures on the line equipped with the Hausdorff metric generated by a suitable variant of $\ell_p$ distance between counting measures.