{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:EUEZ2IRDW7M52ZJKH5YXAV32PM","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":"45fb70f2faaa6b9187f4a794cff206717133b9b3eac645ec1e57595118eed421","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-08T17:43:00Z","title_canon_sha256":"cd3b863c1db7a5529c106d5e5ebe2ab6da8a9b33fac86087e2bcd7b8200e67fe"},"schema_version":"1.0","source":{"id":"1110.1751","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1751","created_at":"2026-05-18T03:36:37Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1751v2","created_at":"2026-05-18T03:36:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1751","created_at":"2026-05-18T03:36:37Z"},{"alias_kind":"pith_short_12","alias_value":"EUEZ2IRDW7M5","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"EUEZ2IRDW7M52ZJK","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"EUEZ2IRD","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:84b4e532c8f0a1c0270a57d7457e35b052d6e3197d6332a3d474bed05e3a46cd","target":"graph","created_at":"2026-05-18T03:36:37Z","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":"The paper deals with the convergence properties of the products of random (row-)stochastic matrices. The limiting behavior of such products is studied from a dynamical system point of view. In particular, by appropriately defining a dynamic associated with a given sequence of random (row-)stochastic matrices, we prove that the dynamics admits a class of time-varying Lyapunov functions, including a quadratic one. Then, we discuss a special class of stochastic matrices, a class $\\Pstar$, which plays a central role in this work. We then introduce balanced chains and using some geometric propertie","authors_text":"Angelia Nedich, Behrouz Touri","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-08T17:43:00Z","title":"Product of Random Stochastic Matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1751","kind":"arxiv","version":2},"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":"sha256:7515b79189e722189430d5b97d2ae669f49476134062edfbaf9cce922ab45a7c","target":"record","created_at":"2026-05-18T03:36:37Z","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":"45fb70f2faaa6b9187f4a794cff206717133b9b3eac645ec1e57595118eed421","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-08T17:43:00Z","title_canon_sha256":"cd3b863c1db7a5529c106d5e5ebe2ab6da8a9b33fac86087e2bcd7b8200e67fe"},"schema_version":"1.0","source":{"id":"1110.1751","kind":"arxiv","version":2}},"canonical_sha256":"25099d2223b7d9dd652a3f7170577a7b39278f61c5bfcbf0183860cdad5698a4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"25099d2223b7d9dd652a3f7170577a7b39278f61c5bfcbf0183860cdad5698a4","first_computed_at":"2026-05-18T03:36:37.111356Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:36:37.111356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9prL/uZ2nyIGZh6PPDMTHPv5C/NIvyckCQFmT7fdfK2lInmTEta2fk2fjcBwHQTxj/+iAV5Ac3+MUVLD/geuAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:36:37.111985Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.1751","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7515b79189e722189430d5b97d2ae669f49476134062edfbaf9cce922ab45a7c","sha256:84b4e532c8f0a1c0270a57d7457e35b052d6e3197d6332a3d474bed05e3a46cd"],"state_sha256":"c568ea622f1eb6f13ce8e5e4ead2d3bdf94898c5ea87a868b6e266a617557580"}