Kac-Stroock type approximations for the Brownian motion
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911278778286080 |
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| author | Bardina, Xavier Boukfal, Salim |
| author_facet | Bardina, Xavier Boukfal, Salim |
| contents | In the present paper we show that the processes $X_n = \{X_n(t) \colon t \in [0,1]\}$, $n \in \mathbb{N}$, defined by $X_n(t) = \sqrt{n}C\int_0^t (-1)^{L(nu)} du$, where $L = \{L(t) \colon t \geq 0\}$ is a renewal processes whose inter-arrival times satisfy some integrability conditions and $C > 0$ is some normalizing constant, weakly converge, in the space of continuous functions over $[0,1]$, $\mathcal{C}([0,1])$, to the Brownian motion as $n$ approaches infinity. Thus, generalizing the result of D. W. Stroock (1982), where $L$ is taken to be a standard Poisson process. In particular, we see that these results are a mere consequence of Donsker's invariance principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kac-Stroock type approximations for the Brownian motion Bardina, Xavier Boukfal, Salim Probability 60F05, 60F17, 60G50, 60K05 In the present paper we show that the processes $X_n = \{X_n(t) \colon t \in [0,1]\}$, $n \in \mathbb{N}$, defined by $X_n(t) = \sqrt{n}C\int_0^t (-1)^{L(nu)} du$, where $L = \{L(t) \colon t \geq 0\}$ is a renewal processes whose inter-arrival times satisfy some integrability conditions and $C > 0$ is some normalizing constant, weakly converge, in the space of continuous functions over $[0,1]$, $\mathcal{C}([0,1])$, to the Brownian motion as $n$ approaches infinity. Thus, generalizing the result of D. W. Stroock (1982), where $L$ is taken to be a standard Poisson process. In particular, we see that these results are a mere consequence of Donsker's invariance principle. |
| title | Kac-Stroock type approximations for the Brownian motion |
| topic | Probability 60F05, 60F17, 60G50, 60K05 |
| url | https://arxiv.org/abs/2511.17281 |