{"paper":{"title":"Stable-like fluctuations of Biggins' martingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Iksanov, Konrad Kolesko, Matthias Meiners","submitted_at":"2017-09-21T15:02:30Z","abstract_excerpt":"Let $(W_n(\\theta))_{n \\in \\mathbb{N}_0}$ be Biggins' martingale associated with a supercritical branching random walk, and let $W(\\theta)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1(\\theta)$ belongs to the domain of normal attraction of an $\\alpha$-stable distribution for some $\\alpha \\in (1,2)$, then, as $n\\to\\infty$, there is weak convergence of the tail process $(W(\\theta) - W_{n-k}(\\theta))_{k \\in \\mathbb{N}_0}$, properly normalized, to a random scale multiple of a stationary autoregressi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.07362","kind":"arxiv","version":1},"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"}