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Specifically, consider an ergodic Markov chain M and a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu = E_{v <- pi}[f(v)], where pi is the stationary distribution of M. A t-step random walk (v_1,...,v_t) on M starting from the stationary distribution pi has expected total weight E[X] = mu t, where X = sum_{i=1}^t f(v_i). Let T be the L_1 mixing-time of M. We show that the probability of X deviating f"},"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":"1201.0559","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-03T01:59:54Z","cross_cats_sorted":[],"title_canon_sha256":"c59793eabc27369d6ae781cd92bef6b5880bbbb3eeaec9f868660256352a25b7","abstract_canon_sha256":"84cf703a57030b6b3487e557abfdc1de80b31bcdf6c64bebb75311b8a23d7851"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:42.564053Z","signature_b64":"kf7B37KP2Y/JbNcAkNKQEJyy5JDhwDFQfHAeE+68FGzv8euNdT5ssPE4fKD8JSXhEmrLCIvPw2lYvkPWeUfNDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5351aa46c50067795408900197d7987802e07947ad11dd8c74ca36f51573300c","last_reissued_at":"2026-05-18T04:03:42.563550Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:42.563550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Chernoff-Hoeffding Bounds for Markov Chains: Generalized and Simplified","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Henry Lam, Kai-Min Chung, Michael Mitzenmacher, Zhenming Liu","submitted_at":"2012-01-03T01:59:54Z","abstract_excerpt":"We prove the first Chernoff-Hoeffding bounds for general nonreversible finite-state Markov chains based on the standard L_1 (variation distance) mixing-time of the chain. Specifically, consider an ergodic Markov chain M and a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu = E_{v <- pi}[f(v)], where pi is the stationary distribution of M. A t-step random walk (v_1,...,v_t) on M starting from the stationary distribution pi has expected total weight E[X] = mu t, where X = sum_{i=1}^t f(v_i). Let T be the L_1 mixing-time of M. 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