{"paper":{"title":"Small deviations of a Galton-Watson process with immigration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Nadia Sidorova","submitted_at":"2015-09-04T15:11:22Z","abstract_excerpt":"We consider a Galton-Watson process with immigration $(\\mathcal{Z}_n)$, with offspring probabilities $(p_i)$ and immigration probabilities $(q_i)$. In the case when $p_0=0$, $p_1\\neq 0$, $q_0=0$ (that is, when $\\text{essinf} (\\mathcal{Z}_n)$ grows linearly in $n$), we establish the asymptotics of the left tail $\\mathbb{P}\\{\\mathcal{W}<\\varepsilon\\}$, as $\\varepsilon\\downarrow 0$, of the martingale limit $\\mathcal{W}$ of the process $( \\mathcal{Z}_n)$. Further, we consider the first generation $\\mathcal{K}$ such that $\\mathcal{Z}_{\\mathcal{K}}>\\text{essinf} (\\mathcal{Z}_{\\mathcal{K}})$ and stud"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01486","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"}