Iterated-logarithm laws for convex hulls of random walks with drift
| Autor: | Cygan, Wojciech, Sandrić, Nikola, Šebek, Stjepan, Wade, Andrew |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Transactions of the American Mathematical Society, Vol. 377 (2024), no. 9, p. 6695-6724 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1090/tran/9238 |
| Popis: | We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our starting point is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero--one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk. Comment: Accepted for publication in TAMS |
| Databáze: | arXiv |
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