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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"},"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":"1509.01486","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-09-04T15:11:22Z","cross_cats_sorted":[],"title_canon_sha256":"59fff9bc84046286946ae6682776603563d9660156cb7cc372e131a2c1feee5b","abstract_canon_sha256":"39c4532ad32651b8ca97df5f58d530f1797cc534308ab6b5e72d36511df49fc2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:14.461630Z","signature_b64":"J9fdOGT4mvIchtu7K2wcH7oQlV4Ars7Sl7WneuRDJWeUX2sxNG0QOn+qCyQXOQX5CwvK+LNxJGarGAPtYKzEBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04b225ec6b93f9a6f23f3a8a8e11f70043c721f644b6db471e085d05c5cd7043","last_reissued_at":"2026-05-18T00:55:14.461138Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:14.461138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $( \\mathcal{Z}_n)$. 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