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In the presence of a non-lattice renewal arrival process and assuming that the service times have a continuous distribution, we obtain a large deviations principle for $Q_{\\lambda}(\\cdot) /\\lambda$ under the topology of uniform convergence on $[0,T]\\times\\lbrack0,\\infty)$. We illustrate our results by obtaining the most likely path, represented as a surface, to ruin in life insurance portfolios, and also we obtain the "},"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":"1207.5164","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-21T17:32:14Z","cross_cats_sorted":[],"title_canon_sha256":"9f6e042d6ad3a9ab543e769e6cfe934d19b464bcbbc42bba8b58c3e1c27f5ff6","abstract_canon_sha256":"212b7f757143fa28af309866011bf85c02efd68b0bf1ae2db2d4084b38852aba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:24.325497Z","signature_b64":"K+3xT5pOpjxzFPNkrVDbJBSDLq1f5waXpd4dilYnRWT1pEt+ly8toM7vZ/rD4yerygYjIaIftupLOo+KuVceAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ca3610841d482e4044716ac936eaf31898772e804614d4b9470746140a76a95","last_reissued_at":"2026-05-18T03:50:24.324757Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:24.324757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two-parameter Sample Path Large Deviations for Infinite Server Queues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"H. 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