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It is shown in this note that, if the dual sequence $({}^{\\#}M_{n},{}^{\\#}S_{n})_{n\\ge 0}$ is positive divergent, i.e. ${}^{\\#}S_{n}\\to\\infty$ a.s., then the strictly ascending ladder epochs $\\sigma_{n}^{>}$ of $(M_{n},S_{n})_{n\\ge 0}$ are a.s. finite and the ladder chain $(M_{\\sigma_{n}^{>}})_{n\\ge 0}$ is positive recurrent on some $\\mathcal{S}^{>}\\subset\\mathcal{S}$. We also provide simple expressions for i"},"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":"1511.05361","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-17T11:56:14Z","cross_cats_sorted":[],"title_canon_sha256":"e11f6077f4974f5c048f99657504c6eabd926eb83829997c983c7496567153c6","abstract_canon_sha256":"98cc5c786fcfba0497a2c84bb63690a3e5278692ca0b2c4649ebffc144ba3fcc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:40.518775Z","signature_b64":"hEOKe0ALah96/+3shc5STG0o+VkbbOgrVpOKQ62s8YIXWFO8O5dsVeIrLUMuM3+CF6gByxs01IteOxY6EubuDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cc79d070eb9e954e3ae786bf2f13eb437d3302659101cd2d5a9c8f677704c801","last_reissued_at":"2026-05-18T01:26:40.518147Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:40.518147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ladder epochs and ladder chain of a Markov random walk with discrete driving chain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gerold Alsmeyer","submitted_at":"2015-11-17T11:56:14Z","abstract_excerpt":"Let $(M_{n},S_{n})_{n\\ge 0}$ be a Markov random walk with positive recurrent driving chain $(M_{n})_{n\\ge 0}$ having countable state space $\\mathcal{S}$ and stationary distribution $\\pi$. It is shown in this note that, if the dual sequence $({}^{\\#}M_{n},{}^{\\#}S_{n})_{n\\ge 0}$ is positive divergent, i.e. ${}^{\\#}S_{n}\\to\\infty$ a.s., then the strictly ascending ladder epochs $\\sigma_{n}^{>}$ of $(M_{n},S_{n})_{n\\ge 0}$ are a.s. finite and the ladder chain $(M_{\\sigma_{n}^{>}})_{n\\ge 0}$ is positive recurrent on some $\\mathcal{S}^{>}\\subset\\mathcal{S}$. 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