{"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). Let $\\widehat\\mu(z)$ be the generating function of the intensity of the offspring process, and consider the complex roots of $\\widehat\\mu(z)=1$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02648","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"}