Ranges of Extremal Processes and Heavy-Tailed Random Walks in Spaces of Growing Dimension

TL;DR

This paper studies the asymptotic behavior of extremal processes and heavy-tailed random walks in high-dimensional spaces, establishing convergence results for their paths as both steps and dimensions grow.

Contribution

It introduces new limit theorems for paths of heavy-tailed processes in high dimensions, identifying their limits as Poisson cluster processes and metric space convergences.

Findings

Paths converge to Poisson cluster processes under suitable transformations.

Finite metric space convergence in Gromov-Hausdorff metric is established.

Convergence of transformed paths as counting measures with Hausdorff metric.

Abstract

We consider extremal processes and random walks generated by heavy-tailed random vectors taking values in $\mathbb{R}^d$ endowed with the $\ell_p$ metric. We establish limit theorems for the associated 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 transform the paths by suitable isometries of $\ell_p$ such that the transformed paths converge in distribution and to identify the limit in terms of a Poisson cluster process. These results also imply the convergence in distribution of the paths viewed as finite metric spaces in the space of metric spaces equipped with the Gromov-Hausdorff metric. Furthermore, we prove convergence in distribution of the transformed paths in the space of counting measures on the line equipped with a Hausdorff metric induced by a suitable $\ell_p$-type distance between counting measures.