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Approximately Jumping Towards the Origin
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Approximately Jumping Towards the Origin
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Paper
Approximately Jumping Towards the Origin
2024
Overview
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.
Publisher
Cornell University Library, arXiv.org
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