{"title":"Sylvester's problem for random walks and bridges","authors":"Hugo Panzo","doi":"arxiv-2409.07927","DOIUrl":null,"url":null,"abstract":"Consider a random walk in $\\mathbb{R}^d$ that starts at the origin and whose\nincrement distribution assigns zero probability to any affine hyperplane. We\nsolve Sylvester's problem for these random walks by showing that the\nprobability that the first $d+2$ steps of the walk are in convex position is\nequal to $1-\\frac{2}{(d+1)!}$. The analogous result also holds for random\nbridges of length $d+2$, so long as the joint increment distribution is\nexchangeable.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"88 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.07927","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Consider a random walk in $\mathbb{R}^d$ that starts at the origin and whose
increment distribution assigns zero probability to any affine hyperplane. We
solve Sylvester's problem for these random walks by showing that the
probability that the first $d+2$ steps of the walk are in convex position is
equal to $1-\frac{2}{(d+1)!}$. The analogous result also holds for random
bridges of length $d+2$, so long as the joint increment distribution is
exchangeable.