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We exhibit a phase transition with the critical value $r_k>0$ determined by the equation $\\mathbb E p_1^k = \\mathbb E m_0^{-r_k},$ where $m_0=\\sum_{k=0}^\\infty k p_k$ with $p_k=\\mathbb P(Z_1=k | \\xi),$ assuming that $p_0=0.$ Contrary to the constant environment case (the Galton-Watson case), this critical value is different from that for the existence "},"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":"1608.08075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-08-29T14:38:34Z","cross_cats_sorted":[],"title_canon_sha256":"7aff78b5c5fee442a7c56659ca800d72957174c191c3b73f60f3b530d786555d","abstract_canon_sha256":"6f834d2b65d2066ebad64f70e83b0a08b7489276f1435f222830c0ee8f0031df"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:48.767079Z","signature_b64":"GD0+NAxSk3aprAbeBqGuOyieN6VM4N0tayxBdjA7rAsxyuMT/ZseYYkg6v8QHYW6Eyb4aXXFXpyd5+uTvf1pAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"468913408cb7da8c1e0dc9e330159a5b15d8e1483988b8b85322bc188dd98549","last_reissued_at":"2026-05-18T01:07:48.766625Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:48.766625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Harmonic moments and large deviations for a supercritical branching process in a random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Eric Miqueu, Ion Grama, Quansheng Liu","submitted_at":"2016-08-29T14:38:34Z","abstract_excerpt":"Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\\xi$. 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