{"paper":{"title":"The size of the last merger and time reversal in $\\Lambda$-coalescents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anton Wakolbinger, G\\\"otz Kersting, Jason Schweinsberg","submitted_at":"2017-01-02T22:48:36Z","abstract_excerpt":"We consider the number of blocks involved in the last merger of a $\\Lambda$-coalescent started with $n$ blocks. We give conditions under which, as $n \\to \\infty$, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for $\\Lambda$-coalescents that have a dust component. For general $\\Lambda$, we relate the three cases to the existence, uniqueness and non-existence of quasi-invariant measures for the dynamics of the block-counting process, and in case b) investigate the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00549","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"}