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We establish here a law of iterated logarithm for the upper limits of $M_n$: upon the system's non-extinction, $ \\limsup\\_{n\\to \\infty} {1\\over \\log \\log \\log n} ( M_n - {3\\over2} \\log n) = 1$ almost surely. We also study the problem of moderate deviations of $M_n$: $p(M_n- {3 \\over 2} \\log n > \\lambda)$ for $\\lambda\\to \\infty$ and $\\lambda=o(\\log n)$. This problem is closely related to the small deviations of a"},"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":"1305.6448","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-05-28T11:22:40Z","cross_cats_sorted":[],"title_canon_sha256":"f821235fc2020f0457eb9b7936310aaf49d9453bc434a287781a3e1e6b682b29","abstract_canon_sha256":"1386939cbd2c95d32b5216f3942885be721bdadd7ae060dc08fe043de5bb5b37"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:54.260195Z","signature_b64":"TROnLYds28wf4Abdn4UNhYI3WHIItQVZwHYOiMK/qha4Aoo+uG0+PfujTSY4GlJ4704AdFnHIDgSEMHs7S68Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94a2dd84edfe2b5f3da05b2d55812afcbd555dafd51c7b5e35376c6932afcb77","last_reissued_at":"2026-05-18T00:40:54.259756Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:54.259756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"How big is the minimum of a branching random walk?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yueyun Hu (LAGA)","submitted_at":"2013-05-28T11:22:40Z","abstract_excerpt":"Let $M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n \\to \\infty$, $M_n- {3 \\over 2} \\log n$ is tight (see [1][9][2]). We establish here a law of iterated logarithm for the upper limits of $M_n$: upon the system's non-extinction, $ \\limsup\\_{n\\to \\infty} {1\\over \\log \\log \\log n} ( M_n - {3\\over2} \\log n) = 1$ almost surely. We also study the problem of moderate deviations of $M_n$: $p(M_n- {3 \\over 2} \\log n > \\lambda)$ for $\\lambda\\to \\infty$ and $\\lambda=o(\\log n)$. 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