| Autor: |
Albors, Alex, Clément, François, Kiami, Shosuke, Sodt, Braeden, Yifan, Ding, Zeng, Tony |
| Rok vydání: |
2024 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
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 $\|x_n\| \sim \sqrt{\pi d/8}$ from the origin. We show that the sequence $(\|x_n\|)_{n=1}^{\infty}$ has an invariant measure $\pi_d$ 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. |
| Databáze: |
arXiv |
| Externí odkaz: |
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