{"paper":{"title":"Convergence of complex martingales in the branching random walk: the boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Konrad Kolesko, Matthias Meiners","submitted_at":"2016-11-16T10:58:41Z","abstract_excerpt":"Biggins [Uniform convergence of martingales in the branching random walk. {\\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $\\lambda$ from an open set $\\Lambda \\subseteq \\mathbb{C}^d$. We investigate the martingales corresponding to parameters from the boundary $\\partial \\Lambda$ of $\\Lambda$. The boundary can be decomposed into several parts. There may be a part of the boundary, on which the martingales do not exist, on other parts it exists, but diverges or vanishes "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05220","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"}