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In this paper, we prove a uniform\n Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(X_n,S_n),n\\geq0}.\n To be more precise, for given b\\geq0, define the stopping time \\tau=\\tau(b)=inf{n:S_n>b}.\n When a drift \\mu of the random walk S_n is 0, we deri"},"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":"math/0407140","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-07-08T14:41:23Z","cross_cats_sorted":[],"title_canon_sha256":"b61ed3df0f571c34c6683b2a51ff31ff71718882dbfee032fab7bbb77d2c6117","abstract_canon_sha256":"b00c4b4951fd3c9dc65b16087ffa6742b59ad1ff580efecb57fd645430593322"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:26.189224Z","signature_b64":"L+UOCsAph3zq0PS6h5C7mvpeyqbDWunytKjENcB5iGZ9UP5T/Vlz91aYlV+yQnWXq7EAKZ2NqndTyNUI/o6dAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"17e06a2f43e76ad37f642211f56f2998f39cc7ae9e06b66ca18601d62712694b","last_reissued_at":"2026-05-18T01:05:26.188697Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:26.188697Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniform Markov Renewal Theory and Ruin Probabilities in Markov Random Walks","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Cheng-Der Fuh","submitted_at":"2004-07-08T14:41:23Z","abstract_excerpt":"Let {X_n,n\\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability \\pi. 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