{"paper":{"title":"Solutions to complex smoothing equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Matthias Meiners, Sebastian Mentemeier","submitted_at":"2015-07-29T07:58:38Z","abstract_excerpt":"We consider smoothing equations of the form $$X ~\\stackrel{\\mathrm{law}}{=}~ \\sum_{j \\geq 1} T_j X_j + C$$ where $(C,T_1,T_2,\\ldots)$ is a given sequence of random variables and $X_1,X_2,\\ldots$ are independent copies of $X$ and independent of the sequence $(C,T_1,T_2,\\ldots)$. The focus is on complex smoothing equations, i.e., the case where the random variables $X, C,T_1,T_2,\\ldots$ are complex-valued, but also more general multivariate smoothing equations are considered, in which the $T_j$ are similarity matrices. Under mild assumptions on $(C,T_1,T_2,\\ldots)$, we describe the laws of all r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08043","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"}