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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.04284 |
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| _version_ | 1866916509238951936 |
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| author | Albors, Alex Clément, François Kiami, Shosuke Sodt, Braeden Yifan, Ding Zeng, Tony |
| author_facet | Albors, Alex Clément, François Kiami, Shosuke Sodt, Braeden Yifan, Ding Zeng, Tony |
| contents | Given an initial point $x_0 \in \mathbb{R}^d$ and a sequence of vectors $v_1, v_2, \dots$ in $\mathbb{R}^d$, we define a greedy sequence by setting $x_{n} = x_{n-1} \pm v_n$ where the sign is chosen so as to minimize $\|x_n\|$. We prove that if the vectors $v_i$ are chosen uniformly at random from $\mathbb{S}^{d-1}$ then elements of the sequence are, on average, approximately at distance $\|x_n\| \sim \sqrt{πd/8}$ from the origin. We show that the sequence $(\|x_n\|)_{n=1}^{\infty}$ has an invariant measure $π_d$ depending only on $d$ and we determine its mean and study its decay for all $d$. We also investigate a completely deterministic example in $d=2$ where the $v_n$ are derived from the van der Corput sequence. Several additional examples are considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04284 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximately Jumping Towards the Origin Albors, Alex Clément, François Kiami, Shosuke Sodt, Braeden Yifan, Ding Zeng, Tony Probability Dynamical Systems Given an initial point $x_0 \in \mathbb{R}^d$ and a sequence of vectors $v_1, v_2, \dots$ in $\mathbb{R}^d$, we define a greedy sequence by setting $x_{n} = x_{n-1} \pm v_n$ where the sign is chosen so as to minimize $\|x_n\|$. We prove that if the vectors $v_i$ are chosen uniformly at random from $\mathbb{S}^{d-1}$ then elements of the sequence are, on average, approximately at distance $\|x_n\| \sim \sqrt{πd/8}$ from the origin. We show that the sequence $(\|x_n\|)_{n=1}^{\infty}$ has an invariant measure $π_d$ depending only on $d$ and we determine its mean and study its decay for all $d$. We also investigate a completely deterministic example in $d=2$ where the $v_n$ are derived from the van der Corput sequence. Several additional examples are considered. |
| title | Approximately Jumping Towards the Origin |
| topic | Probability Dynamical Systems |
| url | https://arxiv.org/abs/2412.04284 |