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(1) Under a moment condition of order $p\\in (1,2)$, $W-W_n = o (e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming only $EW_1 \\log W_1^{\\alpha+1} < \\infty$ for some $\\alpha >0$, we have $W-W_n = o (n^{-\\alpha})$ a.s.; similar conclusions hold for a branching process in a varying environment. (2) Under a second mom"},"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":"1010.6111","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-10-28T23:37:45Z","cross_cats_sorted":[],"title_canon_sha256":"9146c0f6dacfe7a3f1ef8e8806d279beb586b8d2143594de5a79ee381481077f","abstract_canon_sha256":"79c0ac0c60e4fe45fda58d133a59976e226f19294da99fc20a97125c14722f53"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:26.865577Z","signature_b64":"YJHMFCC542fQO65/duZW/8+av3s6m3rMfisxZ4IxZr42CxtRFw07JkzQ+pGVQJpsTaRffGTU1uy+hKw94X3eBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd566e0bafdff534f71636a552d9fcb9349fcb42cb5fc6adb6fe1057f0b9f55d","last_reissued_at":"2026-05-18T03:33:26.864877Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:26.864877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convergence rates for a branching process in a random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chunmao Huang, Quansheng Liu","submitted_at":"2010-10-28T23:37:45Z","abstract_excerpt":"Let $(Z_n)$ be a supercritical branching process in a random environment $\\xi$. We study the convergence rates of the martingale $W_n = Z_n/ E[Z_n| \\xi]$ to its limit $W$. The following results about the convergence almost sur (a.s.), in law or in probability, are shown. (1) Under a moment condition of order $p\\in (1,2)$, $W-W_n = o (e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming only $EW_1 \\log W_1^{\\alpha+1} < \\infty$ for some $\\alpha >0$, we have $W-W_n = o (n^{-\\alpha})$ a.s.; similar conclusions hold for a branching process in a varying environment. 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