Abstract
We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in . We prove convergence of the convex hull in the space of all convex and compact subsets of , equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable Lévy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk.
Funding Statement
This work has been supported by Deutscher Akademischer Austauschdienst (DAAD) and Ministry of Science and Education of the Republic of Croatia (MSE) via project Random Time-Change and Jump Processes. Financial support through the Alexander-von-Humboldt Foundation under project No. HRV 1151902 HFST-E and Croatian Science Foundation under project 8958 (for N. Sandrić), and Croatian Science Foundation under project 4197 (for S. Šebek) is gratefully acknowledged.
Acknowledgments
We thank the anonymous referee for helpful comments that have led to improvements of the presentation of the article. We also thank M. Puljiz for discussions and ideas related to the proof of Theorem 3.11.
Citation
Wojciech Cygan. Nikola Sandrić. Stjepan Šebek. "Convex hulls of stable random walks." Electron. J. Probab. 27 1 - 30, 2022. https://doi.org/10.1214/22-EJP826
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