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For a possibly unbounded function $f:\\stany \\to R,$ let $I=\\int_{\\stany} f(x) \\pi(x) dx$ be the value of interest and $\\hat{I}_{t,n}=(1/n)\\sum_{i=t}^{t+n-1}f(X_i)$ its MCMC estimate. Precisely, we derive lower bounds for the length of the trajectory $n$ and burn-in time $t$ which ensure that $$P(|\\hat{I}_{t,n}-I|\\leq \\varepsilon"},"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":"0908.2098","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ME","submitted_at":"2009-08-14T17:01:15Z","cross_cats_sorted":["stat.CO"],"title_canon_sha256":"7cdf965e698d117f4d9f3be43e0dd22ee1e6453269b05b4103a45a0d070ede9e","abstract_canon_sha256":"b29fed61ddf7b9ea30bb7e288e7dd0abe1e3a1caa5220c60ad01fb280866dc1c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:30:26.549838Z","signature_b64":"vPML9Raj+CP5pZo5NYgNSdub7CKUDVZUa9wl9kx2S1hlFZsxxi98GcN6id8M4MFiKhcVbteH3t+7fGDaAx4pBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7474ef8a11a129ef945c8b6933f24bf15fc0cc5be2d299ff1f7dc48fd740d589","last_reissued_at":"2026-05-18T04:30:26.549434Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:30:26.549434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rigorous confidence bounds for MCMC under a geometric drift condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.CO"],"primary_cat":"stat.ME","authors_text":"Krzysztof Latuszynski, Wojciech Niemiro","submitted_at":"2009-08-14T17:01:15Z","abstract_excerpt":"We assume a drift condition towards a small set and bound the mean square error of estimators obtained by taking averages along a single trajectory of a Markov chain Monte Carlo algorithm. 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