{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/ordinal-pattern-probabilities-for-symmetric","title":"Ordinal pattern probabilities for symmetric random walks","arxiv_id":"1907.07172","date":"2019-07-16","proceeding":null,"authors":["Hugh Denoncourt"],"abstract":"An ordinal pattern for a finite sequence of real numbers is a permutation that records the relative positions in the sequence. For random walks with steps drawn uniformly from $[-1,1]$, we show an ordinal pattern occurs with probability $\\frac{|[1,w]|}{2^n n!}$, where $[1,w]$ is a weak order interval in the affine Weyl group $\\widetilde{A}_n$. For random walks with steps drawn from a symmetric Laplace distribution, the probability is $\\frac{1}{2^n \\prod_{j=1}^n \\mathrm{lev}(\\pi)_j}$, where $\\mathrm{lev}(\\pi)_j$ measures how often $j$ occurs between consecutive values in $\\pi$. Permutations whose consecutive values are at most two positions apart in $\\pi$ are shown to occur with the same probability for any choice of symmetric continuous step distribution. For random walks with steps from a mean zero normal distribution, ordinal pattern probabilities are determined by a matrix whose $ij$-th entry measures how often $i$ and $j$ are between consecutive values.","url_abs":"http://arxiv.org/abs/1907.07172v3","url_pdf":"http://arxiv.org/pdf/1907.07172v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"ordinal-pattern-probabilities-for-symmetric","repo_url":"https://github.com/HughDen/OrdinalPatterns","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}