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It is well known that if $E{\\ln}|M|<0$ and $E{\\ln^+}|Q|<\\infty$, then the sequence $(R_n)$ converges in distribution to a random variable $R$ given by $R\\stackrel{d}{=}\\sum_{k=1}^{\\infty}Q_k\\prod_{j=1}^{k-1}M_j$, and usually referred to as perpetuity. In this paper we consider a situation in which the sequence $(R_n)$ itself does no"},"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":"1107.2753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2011-07-14T08:45:02Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"d93b7694ea4903230a547a676882e29ba6ff93b4dd3c290cb939fc6c33f47f06","abstract_canon_sha256":"4046748c58326de5bd3f295cb2e6b450e3e4c25a2b2726f7b4c857a90bfc7962"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:15.727258Z","signature_b64":"K8uzPRr5VWNZBxC1cb4f0pPBWIzcqGrd7vWOlrwWxXihB9t6w3vDj4hcJ85wqlYN54uu2XgExT3R/gCTwYGhDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"744441dce1ca73e47086ca9a84a1ce915c7941566d76bf485200f02ed3663d01","last_reissued_at":"2026-05-18T04:18:15.726887Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:15.726887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Renorming divergent perpetuities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Jacek Weso{\\l}owski, Pawe{\\l} Hitczenko","submitted_at":"2011-07-14T08:45:02Z","abstract_excerpt":"We consider a sequence of random variables $(R_n)$ defined by the recurrence $R_n=Q_n+M_nR_{n-1}$, $n\\ge1$, where $R_0$ is arbitrary and $(Q_n,M_n)$, $n\\ge1$, are i.i.d. copies of a two-dimensional random vector $(Q,M)$, and $(Q_n,M_n)$ is independent of $R_{n-1}$. It is well known that if $E{\\ln}|M|<0$ and $E{\\ln^+}|Q|<\\infty$, then the sequence $(R_n)$ converges in distribution to a random variable $R$ given by $R\\stackrel{d}{=}\\sum_{k=1}^{\\infty}Q_k\\prod_{j=1}^{k-1}M_j$, and usually referred to as perpetuity. 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