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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1403.6971","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-03-27T10:31:19Z","cross_cats_sorted":[],"title_canon_sha256":"e4d0879dea16260a4358e7b7c1d81fa7513775d24f20f315bcf373fa17f147f4","abstract_canon_sha256":"105c4bed24b28c24eb35a103d0e66ab6d0703e6dd5c31ad100786f9516b87705"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:26.775428Z","signature_b64":"OaIk5g/eu92kkR5osnu+WbJJnpWCEUMQkWaAc/0cPSHE6SgDkrxJ1+Nd6wDgsYtGEbfWs3iw96xwz3ItrnceBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64bf3766a1db05dd15a1ecd8a442f32b0b71491d97551628b275294ea8d77812","last_reissued_at":"2026-05-18T02:55:26.774991Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:26.774991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1403.6971","created_at":"2026-05-18T02:55:26.775056+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.6971v1","created_at":"2026-05-18T02:55:26.775056+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.6971","created_at":"2026-05-18T02:55:26.775056+00:00"},{"alias_kind":"pith_short_12","alias_value":"MS7TOZVB3MC5","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_16","alias_value":"MS7TOZVB3MC52FNB","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_8","alias_value":"MS7TOZVB","created_at":"2026-05-18T12:28:38.356838+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MS7TOZVB3MC52FNB5TMKIQXTFM","json":"https://pith.science/pith/MS7TOZVB3MC52FNB5TMKIQXTFM.json","graph_json":"https://pith.science/api/pith-number/MS7TOZVB3MC52FNB5TMKIQXTFM/graph.json","events_json":"https://pith.science/api/pith-number/MS7TOZVB3MC52FNB5TMKIQXTFM/events.json","paper":"https://pith.science/paper/MS7TOZVB"},"agent_actions":{"view_html":"https://pith.science/pith/MS7TOZVB3MC52FNB5TMKIQXTFM","download_json":"https://pith.science/pith/MS7TOZVB3MC52FNB5TMKIQXTFM.json","view_paper":"https://pith.science/paper/MS7TOZVB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.6971&json=true","fetch_graph":"https://pith.science/api/pith-number/MS7TOZVB3MC52FNB5TMKIQXTFM/graph.json","fetch_events":"https://pith.science/api/pith-number/MS7TOZVB3MC52FNB5TMKIQXTFM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MS7TOZVB3MC52FNB5TMKIQXTFM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MS7TOZVB3MC52FNB5TMKIQXTFM/action/storage_attestation","attest_author":"https://pith.science/pith/MS7TOZVB3MC52FNB5TMKIQXTFM/action/author_attestation","sign_citation":"https://pith.science/pith/MS7TOZVB3MC52FNB5TMKIQXTFM/action/citation_signature","submit_replication":"https://pith.science/pith/MS7TOZVB3MC52FNB5TMKIQXTFM/action/replication_record"}},"created_at":"2026-05-18T02:55:26.775056+00:00","updated_at":"2026-05-18T02:55:26.775056+00:00"}