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Define $S_0:=0$ and $S_k:=\\sum_{i=1}^k f(A_i)$ the successive partial sums, $S^+$ the maximal non-negative partial sum, $Q_1$ the maximal segmental score of the first non-negative excursion and $M_n:=\\max_{0\\leq k\\leq\\ell\\leq n} (S_{\\ell}-S_k)$ the local score first defined by Karlin and Altschul (1990). We establish recursive formulae for the exact distribution of $S^+$ and derive new approximations for the distr"},"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":"1803.02769","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-07T17:02:24Z","cross_cats_sorted":[],"title_canon_sha256":"9ae37fdaa8e07c818e7b6f93f75a11fb08ef2c82925602517983abfa6aacfc4c","abstract_canon_sha256":"a0af5291d143690fc90257f9b32a2b239d57ff154904952141423ac82ab22bb7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:49.286245Z","signature_b64":"YouSm53kgWwb+gFQQtHWOwXEi524SD1XW8uLhOVI4Us9kyE5vrO7csbBWkqXlyLXYdGihD9NxOcZvrYXQwQGBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c777d1a4e424e5c865e40f03343a660ac9d4e0a993597a3be8f146d79b0faf4e","last_reissued_at":"2026-05-18T00:21:49.285673Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:49.285673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improvements on the distribution of maximal segmental scores in a Markovian sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Sabine Mercier, Simona Grusea","submitted_at":"2018-03-07T17:02:24Z","abstract_excerpt":"Let $(A_i)_{i \\geq 0}$ be a finite state irreducible aperiodic Markov chain and $f$ a lattice score function such that the average score is negative and positive scores are possible. Define $S_0:=0$ and $S_k:=\\sum_{i=1}^k f(A_i)$ the successive partial sums, $S^+$ the maximal non-negative partial sum, $Q_1$ the maximal segmental score of the first non-negative excursion and $M_n:=\\max_{0\\leq k\\leq\\ell\\leq n} (S_{\\ell}-S_k)$ the local score first defined by Karlin and Altschul (1990). 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