{"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"}