{"paper":{"title":"The multidimensional truncated Moment Problem: Gaussian and Log-Normal Mixtures, their Carath\\'eodory Numbers, and Set of Atoms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Philipp J. di Dio","submitted_at":"2018-04-19T09:55:58Z","abstract_excerpt":"We study truncated moment sequences of distribution mixtures, especially from Gaussian and log-normal distributions and their Carath\\'eodory numbers. For $\\mathsf{A} = \\{a_1,\\dots,a_m\\}$ continuous (sufficiently differentiable) functions on $\\mathbb{R}^n$ we give a general upper bound of $m-1$ and a general lower bound of $\\left\\lceil \\frac{2m}{(n+1)(n+2)}\\right\\rceil$. For polynomials of degree at most $d$ in $n$ variables we find that the number of Gaussian and log-normal mixtures is bounded by the Carath\\'eodory numbers in \\cite{didio17Cara}. Therefore, for univariate polynomials $\\{1,x,\\do"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.07058","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"}