Scaling Limit of Dependent Random Walk
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918264822562816 |
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| author | Lee, Jeonghwa |
| author_facet | Lee, Jeonghwa |
| contents | Recently, a generalized Bernoulli process (GBP) was developed as a stationary binary sequence that can have long-range dependence. In this paper, we find the scaling limit of a random walk that follows GBP. The result is a new class of non-Markovian diffusion processes. The limiting processes include continuous-time stochastic processes with stationary increments whose correlation decays with an exponential rate, a power law, or an exponentially tempered power law. The limit densities solve a time tempered fractional diffusion equation or time fractional diffusion equation. The second-family of Mittag-Leffler distribution and exponential distribution arise as special cases of the limiting distributions. Subordinated processes are considered as time-changed Lévy processes, and the governing equations and dependence structure of the subordinated processes are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14447 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scaling Limit of Dependent Random Walk Lee, Jeonghwa Probability Recently, a generalized Bernoulli process (GBP) was developed as a stationary binary sequence that can have long-range dependence. In this paper, we find the scaling limit of a random walk that follows GBP. The result is a new class of non-Markovian diffusion processes. The limiting processes include continuous-time stochastic processes with stationary increments whose correlation decays with an exponential rate, a power law, or an exponentially tempered power law. The limit densities solve a time tempered fractional diffusion equation or time fractional diffusion equation. The second-family of Mittag-Leffler distribution and exponential distribution arise as special cases of the limiting distributions. Subordinated processes are considered as time-changed Lévy processes, and the governing equations and dependence structure of the subordinated processes are discussed. |
| title | Scaling Limit of Dependent Random Walk |
| topic | Probability |
| url | https://arxiv.org/abs/2504.14447 |