Ranges of Extremal Processes and Heavy-Tailed Random Walks in Spaces of Growing Dimension
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917461761196032 |
|---|---|
| author | Jin, Bochen Molchanov, Ilya |
| author_facet | Jin, Bochen Molchanov, Ilya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ranges of Extremal Processes and Heavy-Tailed Random Walks in Spaces of Growing Dimension Jin, Bochen Molchanov, Ilya Probability 60B10, 60F05, 60G55, 60G70 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. |
| title | Ranges of Extremal Processes and Heavy-Tailed Random Walks in Spaces of Growing Dimension |
| topic | Probability 60B10, 60F05, 60G55, 60G70 |
| url | https://arxiv.org/abs/2510.18780 |