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We assume that $E |B|^\\nu < \\infty$ for some $\\nu > \\kappa$, and consider two cases (i) $E A^\\kappa = 1$, $E A^\\kappa \\log_+ A = \\infty$; (ii) $E A^\\kappa < 1$, $E A^t = \\infty$ for all $t > \\kappa$. We show that under appropriate additional assumptions on $A$ the asymptotic $P \\{ X > x \\} \\"},"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":"1512.07262","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-22T21:09:49Z","cross_cats_sorted":[],"title_canon_sha256":"6057db6d0610bfb1b3d04a3b1f30ef88eb2b929f3a4074fecbe25642e5140b3b","abstract_canon_sha256":"68d888b292fe612fc9d8bea99ebff368e49091c4bee1da75e6a8e0e74a1f9b65"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:39.953689Z","signature_b64":"toZcJe052VZR1LJaFbOWX/IrPyEZNm3hh5lTTIbnEI3IiASh9r6yQSslhSMhDxIP68qKIIi8msQrM0AWTIHXAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"079c71b56208f3b74dd68227b9c9cf9a6b64935c290085310c0a3d99a714473d","last_reissued_at":"2026-05-18T01:10:39.953261Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:39.953261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the Kesten--Grincevi\\v{c}ius--Goldie theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Peter Kevei","submitted_at":"2015-12-22T21:09:49Z","abstract_excerpt":"Consider the perpetuity equation $X \\stackrel{\\mathcal{D}}{=} A X + B$, where $(A,B)$ and $X$ on the right-hand side are independent. The Kesten--Grincevi\\v{c}ius--Goldie theorem states that $P \\{ X > x \\} \\sim c x^{-\\kappa}$ if $E A^\\kappa = 1$, $E A^\\kappa \\log_+ A < \\infty$, and $E |B|^\\kappa < \\infty$. We assume that $E |B|^\\nu < \\infty$ for some $\\nu > \\kappa$, and consider two cases (i) $E A^\\kappa = 1$, $E A^\\kappa \\log_+ A = \\infty$; (ii) $E A^\\kappa < 1$, $E A^t = \\infty$ for all $t > \\kappa$. We show that under appropriate additional assumptions on $A$ the asymptotic $P \\{ X > x \\} \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07262","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1512.07262","created_at":"2026-05-18T01:10:39.953326+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.07262v2","created_at":"2026-05-18T01:10:39.953326+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07262","created_at":"2026-05-18T01:10:39.953326+00:00"},{"alias_kind":"pith_short_12","alias_value":"A6OHDNLCBDZ3","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"A6OHDNLCBDZ3OTOW","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"A6OHDNLC","created_at":"2026-05-18T12:29:10.953037+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/A6OHDNLCBDZ3OTOWQIT3TSOPTJ","json":"https://pith.science/pith/A6OHDNLCBDZ3OTOWQIT3TSOPTJ.json","graph_json":"https://pith.science/api/pith-number/A6OHDNLCBDZ3OTOWQIT3TSOPTJ/graph.json","events_json":"https://pith.science/api/pith-number/A6OHDNLCBDZ3OTOWQIT3TSOPTJ/events.json","paper":"https://pith.science/paper/A6OHDNLC"},"agent_actions":{"view_html":"https://pith.science/pith/A6OHDNLCBDZ3OTOWQIT3TSOPTJ","download_json":"https://pith.science/pith/A6OHDNLCBDZ3OTOWQIT3TSOPTJ.json","view_paper":"https://pith.science/paper/A6OHDNLC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.07262&json=true","fetch_graph":"https://pith.science/api/pith-number/A6OHDNLCBDZ3OTOWQIT3TSOPTJ/graph.json","fetch_events":"https://pith.science/api/pith-number/A6OHDNLCBDZ3OTOWQIT3TSOPTJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A6OHDNLCBDZ3OTOWQIT3TSOPTJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A6OHDNLCBDZ3OTOWQIT3TSOPTJ/action/storage_attestation","attest_author":"https://pith.science/pith/A6OHDNLCBDZ3OTOWQIT3TSOPTJ/action/author_attestation","sign_citation":"https://pith.science/pith/A6OHDNLCBDZ3OTOWQIT3TSOPTJ/action/citation_signature","submit_replication":"https://pith.science/pith/A6OHDNLCBDZ3OTOWQIT3TSOPTJ/action/replication_record"}},"created_at":"2026-05-18T01:10:39.953326+00:00","updated_at":"2026-05-18T01:10:39.953326+00:00"}