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

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.