{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:6X3XZCGZHON7HYV2KGSORDE4GR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c05c37df4fa1270386a8a1b442e73ec3da23f623a737912f624d11559e49e8eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-10-09T17:32:41Z","title_canon_sha256":"157971bc232fa509eecd0d79987f9e967f04e7c5507ea23420daf857553c03eb"},"schema_version":"1.0","source":{"id":"1610.02700","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.02700","created_at":"2026-05-18T01:02:50Z"},{"alias_kind":"arxiv_version","alias_value":"1610.02700v1","created_at":"2026-05-18T01:02:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.02700","created_at":"2026-05-18T01:02:50Z"},{"alias_kind":"pith_short_12","alias_value":"6X3XZCGZHON7","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"6X3XZCGZHON7HYV2","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"6X3XZCGZ","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:5df402c1e0a2b92b8eaa86071e461d0471200f4b41ff9a2b455551736e076868","target":"graph","created_at":"2026-05-18T01:02:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This note investigates invariance principles for sums of N(nt) iid radom variables, where n is an integer, t is a positive real number and N(u) is a stochastic process with nonnegative integer values. We show that the sequence of sums of these random variables denoted S(n,t), when appropriately centered and normalized, weakly converges to a Gaussian process. We give sufficient conditions depending on the expectation of N(nt) which allows to rescale S(n,t) into a stochastic S(n,a(t)) weakly converging to a Brownian motion.","authors_text":"Gane Samb Lo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-10-09T17:32:41Z","title":"Invariance principles for random sums of random variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02700","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f55ff701df09383c77d117e7154661d833b8fc6f2f27b7f5ce50385d72165a09","target":"record","created_at":"2026-05-18T01:02:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c05c37df4fa1270386a8a1b442e73ec3da23f623a737912f624d11559e49e8eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-10-09T17:32:41Z","title_canon_sha256":"157971bc232fa509eecd0d79987f9e967f04e7c5507ea23420daf857553c03eb"},"schema_version":"1.0","source":{"id":"1610.02700","kind":"arxiv","version":1}},"canonical_sha256":"f5f77c88d93b9bf3e2ba51a4e88c9c345c1c5b8d1c22ee5eb773eefb1da08f0b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f5f77c88d93b9bf3e2ba51a4e88c9c345c1c5b8d1c22ee5eb773eefb1da08f0b","first_computed_at":"2026-05-18T01:02:50.913114Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:02:50.913114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1PyHg+Jn9bdQmPHoD/ne0DScZF9CeF3cFJ7zIbqM9fyNkYpavAIILrTozOW9pDI7ZISf5rOJKXZtZm7BblviCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:02:50.913546Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.02700","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f55ff701df09383c77d117e7154661d833b8fc6f2f27b7f5ce50385d72165a09","sha256:5df402c1e0a2b92b8eaa86071e461d0471200f4b41ff9a2b455551736e076868"],"state_sha256":"e37dd13d6c53f702ebdb4a2c242410b372f3b8c643c5b597696d9489da7541e4"}