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The smallest (in absolute value) such root is $e^{-\\alpha}$, where $\\alpha>0$ is the Malthusian parameter; let $\\gamma_*$ be the second smallest absolute value of a root. We show, assuming some technical conditions, that there are three cases:\n  (i) if $\\gamma_*>e^{-\\alpha/2}$, then the second-order fluctuations of the age distribution ar"},"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":"1712.02648","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-06T09:32:01Z","cross_cats_sorted":[],"title_canon_sha256":"f83b940aac00b930ac3e11f6b8a13c1192d3c7d9aba600fd5ed1375b2ca0e060","abstract_canon_sha256":"081c487bcac7b5fa854e6162b774fe169b07917f1e90311780e93fb9856f9222"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:46.832698Z","signature_b64":"+7dsSBu6K62rNwlyOp0VzLK3gJJOd2jQ3jYXMeGJPLq5DfOWxCvQvu5Hs+AjSeBhOMtE/awYkbbHaFPU9I8lDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66040ffea50d07b15f9b02b02d29350aa7f4e7c329482c7a18cbe690c7ebc013","last_reissued_at":"2026-05-18T00:14:46.832061Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:46.832061Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics of fluctuations in Crump-Mode-Jagers processes: the lattice case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Svante Janson","submitted_at":"2017-12-06T09:32:01Z","abstract_excerpt":"Consider a supercritical Crump--Mode--Jagers process such that all births are at integer times (the lattice case). 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