{"paper":{"title":"The scaling limit of Poisson-driven order statistics with applications in geometric probability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Christoph Thaele, Matthias Schulte","submitted_at":"2012-01-25T14:35:19Z","abstract_excerpt":"Let $\\eta_t$ be a Poisson point process of intensity $t\\geq 1$ on some state space $\\Y$ and $f$ be a non-negative symmetric function on $\\Y^k$ for some $k\\geq 1$. Applying $f$ to all $k$-tuples of distinct points of $\\eta_t$ generates a point process $\\xi_t$ on the positive real-half axis. The scaling limit of $\\xi_t$ as $t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the the $m$-th smallest point of $\\xi_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background include"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5282","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}