{"paper":{"title":"Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"J. Kilian, R. Arratia, S. Garibaldi","submitted_at":"2013-10-26T00:01:39Z","abstract_excerpt":"We study the random variable B(c,n), which counts the number of balls that must be thrown into n equally-sized bins in order to obtain c collisions. The asymptotic expected value of B(1,n) is the well-known $\\sqrt{n\\pi/2}$ appearing in the solution to the birthday problem; the limit distribution and asymptotic moments of B(1,n) are also well known. We calculate the distribution and moments of B(c,n) asymptotically as n goes to infinity and c = O(n).\n  Our main tools are an embedding of the collision process, realizing the process as a deterministic function of the standard Poisson process, and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7055","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"}