{"paper":{"title":"Fixed points of the smoothing transform: Two-sided solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gerold Alsmeyer, Matthias Meiners","submitted_at":"2010-09-13T15:05:37Z","abstract_excerpt":"Given a sequence $(C,T) = (C,T_1,T_2,...)$ of real-valued random variables with $T_j \\geq 0$ for all $j \\geq 1$ and almost surely finite $N = \\sup\\{j \\geq 1: T_j > 0\\}$, the smoothing transform associated with $(C,T)$, defined on the set $\\mathcal{P}(\\R)$ of probability distributions on the real line, maps an element $P\\in\\mathcal{P}(\\R)$ to the law of $C + \\sum_{j \\geq 1} T_j X_j$, where $X_1,X_2,...$ is a sequence of i.i.d.\\ random variables independent of $(C,T)$ and with distribution $P$. We study the fixed points of the smoothing transform, that is, the solutions to the stochastic fixed-p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2412","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"}