Local times of deterministic paths and self-similar processes with stationary increments as normalized numbers of interval crossings
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910516937490432 |
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| author | Bednorz, Witold Das, Purba Łochowski, Rafał |
| author_facet | Bednorz, Witold Das, Purba Łochowski, Rafał |
| contents | We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a càdlàg path and an occupation measure associated with this path.
Using this result we define local times of fractional Brownian motions (classically defined as densities of relevant occupation measure) as weak limits of properly normalized numbers of interval crossings.
We also discuss a similar result for càdlàg semimartingales, in particular for alpha-stable processes, and for Rosenblatt processes, and provide natural examples of deterministic paths which possess quadratic or higher order variation but no local times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05570 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local times of deterministic paths and self-similar processes with stationary increments as normalized numbers of interval crossings Bednorz, Witold Das, Purba Łochowski, Rafał Probability Functional Analysis 60G17, 60G18, 26A45, 26A30 We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a càdlàg path and an occupation measure associated with this path. Using this result we define local times of fractional Brownian motions (classically defined as densities of relevant occupation measure) as weak limits of properly normalized numbers of interval crossings. We also discuss a similar result for càdlàg semimartingales, in particular for alpha-stable processes, and for Rosenblatt processes, and provide natural examples of deterministic paths which possess quadratic or higher order variation but no local times. |
| title | Local times of deterministic paths and self-similar processes with stationary increments as normalized numbers of interval crossings |
| topic | Probability Functional Analysis 60G17, 60G18, 26A45, 26A30 |
| url | https://arxiv.org/abs/2312.05570 |