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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1701.06294","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-01-23T08:36:37Z","cross_cats_sorted":[],"title_canon_sha256":"1432ab08a5a10740de7dc2443f628f16b8c33377607355be87e426c794dd46bc","abstract_canon_sha256":"d1a7fc4b8be1970cea212931ea0f63d0c6fddac978c7d88726090bd156c223f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:18.738850Z","signature_b64":"5K+72LtMO9mjIysx/hA0f191b420+okk4hlczxsW4Ka72QVrnx0UpcyHYrueJX6I5FBmXrJMR5BYd6R08y52CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc910e684527e434d80d11f58985db885a40097b78aa3b3b071916c176114b8f","last_reissued_at":"2026-05-18T00:52:18.738086Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:18.738086Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $G_i$. 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