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Firstly, the asymptotic distributions of $U^{(n)}$ and $W^{(n)}$ are given. Simultaneously, the errors are estimated by using Chen-Stein method. Next, the almost surely limits are discussed when all $p_n$ are equal and when considered on a common probability space. Finally, we consider the case that $"},"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":"1204.1149","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-04-05T08:41:49Z","cross_cats_sorted":[],"title_canon_sha256":"23cda572261a7b5893933fbc64c99b1774e79605ea437e2fd6171c364bbfe35c","abstract_canon_sha256":"72e119b3ff9aa7cf78df4cbd90f318f03e57089d6822cdde8e272e9307777a19"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:36.238595Z","signature_b64":"pIp8GoY4QBo2NTULlf3ZnzczIItVmzLKKfedsBGFX+tSOyzALScjCXcDm7hqAE8l+1WnFcm457+sCuA/lb8WDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbca44284b36334e1d8a462c38414e9a4c5f507917f5df3d0a190f2113e736f5","last_reissued_at":"2026-05-18T03:58:36.238032Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:36.238032Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the longest length of arithmetic progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Huizeng Zhang, Minzhi Zhao","submitted_at":"2012-04-05T08:41:49Z","abstract_excerpt":"Suppose that $\\xi^{(n)}_1,\\xi^{(n)}_2,...,\\xi^{(n)}_n$ are i.i.d with $P(\\xi^{(n)}_i=1)=p_n=1-P(\\xi^{(n)}_i=0)$. Let $U^{(n)}$ and $W^{(n)}$ be the longest length of arithmetic progressions and of arithmetic progressions mod $n$ relative to $\\xi^{(n)}_1,\\xi^{(n)}_2,..., \\xi^{(n)}_n$ respectively. Firstly, the asymptotic distributions of $U^{(n)}$ and $W^{(n)}$ are given. Simultaneously, the errors are estimated by using Chen-Stein method. Next, the almost surely limits are discussed when all $p_n$ are equal and when considered on a common probability space. 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