{"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/a-correlation-inequality-for-random-points-in","title":"A correlation inequality for random points in a hypercube with some implications","arxiv_id":"2209.00346","date":"2022-09-01","proceeding":null,"authors":["Royi Jacobovic","Or Zuk"],"abstract":"Let $\\prec$ be the product order on $\\mathbb{R}^k$ and assume that $X_1,X_2,\\ldots,X_n$ ($n\\geq3$) are i.i.d. random vectors distributed uniformly in the unit hypercube $[0,1]^k$. Let $S$ be the (random) set of vectors in $\\mathbb{R}^k$ that $\\prec$-dominate all vectors in $\\{X_3,..,X_n\\}$, and let $W$ be the set of vectors that are not $\\prec$-dominated by any vector in $\\{X_3,..,X_n\\}$. The main result of this work is the correlation inequality \\begin{equation*} P(X_2\\in W|X_1\\in W)\\leq P(X_2\\in W|X_1\\in S)\\,. \\end{equation*} For every $1\\leq i \\leq n$ let $E_{i,n}$ be the event that $X_i$ is not $\\prec$-dominated by any of the other vectors in $\\{X_1,\\ldots,X_n\\}$. The main inequality yields an elementary proof for the result that the events $E_{1,n}$ and $E_{2,n}$ are asymptotically independent as $n\\to\\infty$. Furthermore, we derive a related combinatorial formula for the variance of the sum $\\sum_{i=1}^n \\textbf{1}_{E_{i,n}}$, i.e. the number of maxima under the product order $\\prec$, and show that certain linear functionals of partial sums of $\\{\\textbf{1}_{E_{i,n}};1\\leq i\\leq n\\}$ are asymptotically normal as $n\\to\\infty$.","url_abs":"https://arxiv.org/abs/2209.00346v1","url_pdf":"https://arxiv.org/pdf/2209.00346v1.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":"a-correlation-inequality-for-random-points-in","repo_url":"https://github.com/orzuk/pareto","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}