{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:WQOHHLZ2574VWYIHQIOGMGRSEA","short_pith_number":"pith:WQOHHLZ2","schema_version":"1.0","canonical_sha256":"b41c73af3aeff95b6107821c661a322031a274245b0118128a2bb0adaaf4d740","source":{"kind":"arxiv","id":"1906.07260","version":1},"attestation_state":"computed","paper":{"title":"Concentration of Markov chains with bounded moments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Assaf Naor, Oded Regev, Shravas Rao","submitted_at":"2019-06-17T20:51:24Z","abstract_excerpt":"Let $\\{W_t\\}_{t=1}^{\\infty}$ be a finite state stationary Markov chain, and suppose that $f$ is a real-valued function on the state space. If $f$ is bounded, then Gillman's expander Chernoff bound (1993) provides concentration estimates for the random variable $f(W_1)+\\cdots+f(W_n)$ that depend on the spectral gap of the Markov chain and the assumed bound on $f$. Here we obtain analogous inequalities assuming only that the $q$'th moment of $f$ is bounded for some $q \\geq 2$. Our proof relies on reasoning that differs substantially from the proofs of Gillman's theorem that are available in the "},"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":"1906.07260","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-06-17T20:51:24Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"3399c58aec03e93c269433f2ba5d0a99c1db44c0e71c61cdffca46d582414a35","abstract_canon_sha256":"0895ac4ae6c53b2bd9a0d1d9c88ed6600bfba7347bf152003ff0f3b023fc55f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:08.384681Z","signature_b64":"DQBx1iqgo+30lynyoOqK5yEP+e3sErQyqVDxtFlZFqETMU6Ttx4mmop566DeRd2+ANVCKLuxBE7lZDsd2bifCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b41c73af3aeff95b6107821c661a322031a274245b0118128a2bb0adaaf4d740","last_reissued_at":"2026-05-17T23:43:08.384279Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:08.384279Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Concentration of Markov chains with bounded moments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Assaf Naor, Oded Regev, Shravas Rao","submitted_at":"2019-06-17T20:51:24Z","abstract_excerpt":"Let $\\{W_t\\}_{t=1}^{\\infty}$ be a finite state stationary Markov chain, and suppose that $f$ is a real-valued function on the state space. If $f$ is bounded, then Gillman's expander Chernoff bound (1993) provides concentration estimates for the random variable $f(W_1)+\\cdots+f(W_n)$ that depend on the spectral gap of the Markov chain and the assumed bound on $f$. Here we obtain analogous inequalities assuming only that the $q$'th moment of $f$ is bounded for some $q \\geq 2$. Our proof relies on reasoning that differs substantially from the proofs of Gillman's theorem that are available in the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.07260","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":"1906.07260","created_at":"2026-05-17T23:43:08.384345+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.07260v1","created_at":"2026-05-17T23:43:08.384345+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.07260","created_at":"2026-05-17T23:43:08.384345+00:00"},{"alias_kind":"pith_short_12","alias_value":"WQOHHLZ2574V","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_16","alias_value":"WQOHHLZ2574VWYIH","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_8","alias_value":"WQOHHLZ2","created_at":"2026-05-18T12:33:30.264802+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/WQOHHLZ2574VWYIHQIOGMGRSEA","json":"https://pith.science/pith/WQOHHLZ2574VWYIHQIOGMGRSEA.json","graph_json":"https://pith.science/api/pith-number/WQOHHLZ2574VWYIHQIOGMGRSEA/graph.json","events_json":"https://pith.science/api/pith-number/WQOHHLZ2574VWYIHQIOGMGRSEA/events.json","paper":"https://pith.science/paper/WQOHHLZ2"},"agent_actions":{"view_html":"https://pith.science/pith/WQOHHLZ2574VWYIHQIOGMGRSEA","download_json":"https://pith.science/pith/WQOHHLZ2574VWYIHQIOGMGRSEA.json","view_paper":"https://pith.science/paper/WQOHHLZ2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.07260&json=true","fetch_graph":"https://pith.science/api/pith-number/WQOHHLZ2574VWYIHQIOGMGRSEA/graph.json","fetch_events":"https://pith.science/api/pith-number/WQOHHLZ2574VWYIHQIOGMGRSEA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WQOHHLZ2574VWYIHQIOGMGRSEA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WQOHHLZ2574VWYIHQIOGMGRSEA/action/storage_attestation","attest_author":"https://pith.science/pith/WQOHHLZ2574VWYIHQIOGMGRSEA/action/author_attestation","sign_citation":"https://pith.science/pith/WQOHHLZ2574VWYIHQIOGMGRSEA/action/citation_signature","submit_replication":"https://pith.science/pith/WQOHHLZ2574VWYIHQIOGMGRSEA/action/replication_record"}},"created_at":"2026-05-17T23:43:08.384345+00:00","updated_at":"2026-05-17T23:43:08.384345+00:00"}