An almost sure invariance principle for the Takagi-van der Waerden class functions
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911708474245120 |
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| author | Nakano, Yuzaburo |
| author_facet | Nakano, Yuzaburo |
| contents | The Takagi-van der Waerden functions are a well-known class of continuous but nowhere differentiable functions. In this paper, we study their weighted versions, the Takagi-van der Waerden class functions $f_{r,a}(x)$, from a probabilistic point of view. We prove that the local modulus of continuity of $f_{r,a}(x)$ is described by a standard Brownian motion under some regularity assumptions on the weights, as an application of a strong approximation for elephant random walks remembering the very recent past with variable step length. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23461 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An almost sure invariance principle for the Takagi-van der Waerden class functions Nakano, Yuzaburo Probability The Takagi-van der Waerden functions are a well-known class of continuous but nowhere differentiable functions. In this paper, we study their weighted versions, the Takagi-van der Waerden class functions $f_{r,a}(x)$, from a probabilistic point of view. We prove that the local modulus of continuity of $f_{r,a}(x)$ is described by a standard Brownian motion under some regularity assumptions on the weights, as an application of a strong approximation for elephant random walks remembering the very recent past with variable step length. |
| title | An almost sure invariance principle for the Takagi-van der Waerden class functions |
| topic | Probability |
| url | https://arxiv.org/abs/2605.23461 |