Occupation times and areas derived from random sampling
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917694128783360 |
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| author | Aurzada, Frank Döring, Leif Pitters, Helmut H. |
| author_facet | Aurzada, Frank Döring, Leif Pitters, Helmut H. |
| contents | We consider the occupation area of spherical (fractional) Brownian motion, i.e. the area where the process is positive, and show that it is uniformly distributed. For the proof, we introduce a new simple combinatorial view on occupation times of stochastic processes that turns out to be surprisingly effective. A sampling method is used to relate the moments of occupation times to persistence probabilities of random walks that again relate to combinatorial factors in the moments of beta distributions. Our approach also yields a new and completely elementary proof of Lévy's second arcsine law for Brownian motion. Further, combined with Spitzer's formula and the use of Bell polynomials, we give a characterisation of the distribution of the occupation times for all Lévy processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09886 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Occupation times and areas derived from random sampling Aurzada, Frank Döring, Leif Pitters, Helmut H. Probability 60G22, 60G51 (Primary), 60G50 (Secondary) We consider the occupation area of spherical (fractional) Brownian motion, i.e. the area where the process is positive, and show that it is uniformly distributed. For the proof, we introduce a new simple combinatorial view on occupation times of stochastic processes that turns out to be surprisingly effective. A sampling method is used to relate the moments of occupation times to persistence probabilities of random walks that again relate to combinatorial factors in the moments of beta distributions. Our approach also yields a new and completely elementary proof of Lévy's second arcsine law for Brownian motion. Further, combined with Spitzer's formula and the use of Bell polynomials, we give a characterisation of the distribution of the occupation times for all Lévy processes. |
| title | Occupation times and areas derived from random sampling |
| topic | Probability 60G22, 60G51 (Primary), 60G50 (Secondary) |
| url | https://arxiv.org/abs/2406.09886 |