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Assuming essentially that the martingale $(W_n(2\\theta))_{n\\in\\mathbb N_0}$ is uniformly integrable and that $\\text{Var} W_1(\\theta)$ is finite, we prove a functional central limit theorem for the tail process $(W_\\infty(\\theta) - W_{n+r}(\\theta))_{r\\in\\mathbb N_0}$ and a law of the iterated logarithm for $W_\\infty(\\theta)-W_n(\\theta)$, as $n\\to\\infty$."},"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":"1507.08458","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-30T11:21:44Z","cross_cats_sorted":[],"title_canon_sha256":"f270dcf1e57b7b9d1f4c0d7385f796d0ea5061e5089d05f26b081cc0c3d76e02","abstract_canon_sha256":"6daebe8d438fdc792cf1ecde9d2c37ed2367aa2b7802f2e62a76efdb30d62edc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:57.646533Z","signature_b64":"BQFVTqmDouZRq7WNSSaT3Wr2vdzlqLOs3OrAGq9Fde0HmneY/xCjpRBcNe4vUHBZ0ZDyN8nbcjKMlTzDH1hNDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"181e48094b530a248481779d694e557e10a747689d835cf14e9493ca7ad9b02e","last_reissued_at":"2026-05-18T01:22:57.645866Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:57.645866Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Iksanov, Zakhar Kabluchko","submitted_at":"2015-07-30T11:21:44Z","abstract_excerpt":"Let $(W_n(\\theta))_{n\\in\\mathbb N_0}$ be the Biggins martingale associated with a supercritical branching random walk and denote by $W_\\infty(\\theta)$ its limit. Assuming essentially that the martingale $(W_n(2\\theta))_{n\\in\\mathbb N_0}$ is uniformly integrable and that $\\text{Var} W_1(\\theta)$ is finite, we prove a functional central limit theorem for the tail process $(W_\\infty(\\theta) - W_{n+r}(\\theta))_{r\\in\\mathbb N_0}$ and a law of the iterated logarithm for $W_\\infty(\\theta)-W_n(\\theta)$, as $n\\to\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08458","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1507.08458","created_at":"2026-05-18T01:22:57.645976+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.08458v2","created_at":"2026-05-18T01:22:57.645976+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08458","created_at":"2026-05-18T01:22:57.645976+00:00"},{"alias_kind":"pith_short_12","alias_value":"DAPEQCKLKMFC","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"DAPEQCKLKMFCJBEB","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"DAPEQCKL","created_at":"2026-05-18T12:29:17.054201+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DAPEQCKLKMFCJBEBO6OWSTSVPY","json":"https://pith.science/pith/DAPEQCKLKMFCJBEBO6OWSTSVPY.json","graph_json":"https://pith.science/api/pith-number/DAPEQCKLKMFCJBEBO6OWSTSVPY/graph.json","events_json":"https://pith.science/api/pith-number/DAPEQCKLKMFCJBEBO6OWSTSVPY/events.json","paper":"https://pith.science/paper/DAPEQCKL"},"agent_actions":{"view_html":"https://pith.science/pith/DAPEQCKLKMFCJBEBO6OWSTSVPY","download_json":"https://pith.science/pith/DAPEQCKLKMFCJBEBO6OWSTSVPY.json","view_paper":"https://pith.science/paper/DAPEQCKL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.08458&json=true","fetch_graph":"https://pith.science/api/pith-number/DAPEQCKLKMFCJBEBO6OWSTSVPY/graph.json","fetch_events":"https://pith.science/api/pith-number/DAPEQCKLKMFCJBEBO6OWSTSVPY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DAPEQCKLKMFCJBEBO6OWSTSVPY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DAPEQCKLKMFCJBEBO6OWSTSVPY/action/storage_attestation","attest_author":"https://pith.science/pith/DAPEQCKLKMFCJBEBO6OWSTSVPY/action/author_attestation","sign_citation":"https://pith.science/pith/DAPEQCKLKMFCJBEBO6OWSTSVPY/action/citation_signature","submit_replication":"https://pith.science/pith/DAPEQCKLKMFCJBEBO6OWSTSVPY/action/replication_record"}},"created_at":"2026-05-18T01:22:57.645976+00:00","updated_at":"2026-05-18T01:22:57.645976+00:00"}