{"paper":{"title":"Extremes and gaps in sampling from a GEM random discrete distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jim Pitman, Yuri Yakubovich","submitted_at":"2017-01-23T08:36:37Z","abstract_excerpt":"We show that in a sample of size $n$ from a GEM$(0,\\theta)$ random discrete distribution, the gaps $G_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \\le \\cdots \\le X_{n:n}$ of the sample, with the convention $G_{n:n} := X_{1:n} - 1$, are distributed like the first $n$ terms of an infinite sequence of independent geometric$(i/(i+\\theta))$ variables $G_i$. This extends a known result for the minimum $X_{1:n}$ to other gaps in the range of the sample, and implies that the maximum $X_{n:n}$ has the distribution of $1 + \\sum_{i=1}^n G_i$, hence the known result that $X_{n:n}$ gr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.06294","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"}