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We show that under mild and natural conditions on $M$ the suitably normalized extremes of $(R_n)$ converge in distribution to a double exponential random variable. This partially complements a result of de Haan, Resnick, Rootz\\'en, and de Vries who considered extremes of the sequence $(R_n)$ under the assumption that $\\P(M>1)>0$."},"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":"1106.4281","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-06-21T18:02:45Z","cross_cats_sorted":[],"title_canon_sha256":"665026fee0787e2749044949e21afd7fd9dd268353f678f4f4740c220fbdc1e4","abstract_canon_sha256":"4edf6d4878f58dbef8dc6245cee4cc50a1825f3697661b13a3947b4e4e7f9f9c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:33.791189Z","signature_b64":"Kih6GsOQwWIRtsVCKubEPhv5LkrJD69G9a06q07YXFaya0vN8O+y7zx0uRyNRK9qRxyI8lpUHhv3CP/153zGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c9600129667e2ac445f6bcbb81945ce94347c63fa8e34e02b8ac5404b9b3a51","last_reissued_at":"2026-05-18T04:19:33.790616Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:33.790616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convergence to type I distribution of the extremes of sequences defined by random difference equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Pawel Hitczenko","submitted_at":"2011-06-21T18:02:45Z","abstract_excerpt":"We study the extremes of a sequence of random variables $(R_n)$ defined by the recurrence $R_n=M_nR_{n-1}+q$, $n\\ge1$, where $R_0$ is arbitrary, $(M_n)$ are iid copies of a non--degenerate random variable $M$, $0\\le M\\le1$, and $q>0$ is a constant. We show that under mild and natural conditions on $M$ the suitably normalized extremes of $(R_n)$ converge in distribution to a double exponential random variable. 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