{"paper":{"title":"Cluster sets for partial sums and partial sum processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jim Kuelbs, Uwe Einmahl","submitted_at":"2014-03-27T10:31:19Z","abstract_excerpt":"Let $X,X_1,X_2,\\ldots$ be i.i.d. mean zero random vectors with values in a separable Banach space $B$, $S_n=X_1+\\cdots+X_n$ for $n\\ge1$, and assume $\\{c_n:n\\ge1\\}$ is a suitably regular sequence of constants. Furthermore, let $S_{(n)}(t),0\\le t\\le1$ be the corresponding linearly interpolated partial sum processes. We study the cluster sets $A=C(\\{S_n/c_n\\})$ and $\\mathcal{A}=C(\\{S_{(n)}(\\cdot)/c_n\\})$. In particular, $A$ and $\\mathcal{A}$ are shown to be nonrandom, and we derive criteria when elements in $B$ and continuous functions $f:[0,1]\\to B$ belong to $A$ and $\\mathcal{A}$, respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.6971","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"}