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We study birth and death process $N_t$ which, given the environment $\\omega,$ waits at a state $n$ an exponentially distributed time with parameter $\\lambda_n+\\sum_{l=1}^L\\mu^{l}_n$ and then jumps to $n-i$ with probability ${\\mu^i_n}/(\\lambda_n+\\sum_{l=1}^L\\mu^{l}_n),$ $i=1,...,L$ or to $n+1$ with probability ${\\lambda_n}/(\\lambda_n+\\sum_{l=1}^L\\mu^{l}_n).$ A sufficient condition for t"},"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":"1407.3385","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-07-12T13:42:41Z","cross_cats_sorted":[],"title_canon_sha256":"3e17c8e817dc260a8a1593564e6c304294d54760e8a96d147a4165ecea937a6f","abstract_canon_sha256":"dc292e44df174f3b24f1132a30b86d3a26a0e0b17991b33a26b521b23c06da68"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:43.068662Z","signature_b64":"RgzcBRo+8xHijoSrVacSaloqUcxRdzqjko3Dr38jNfX3ZUdwGBYSL4fzApBSd4vdmMdvTNxtn1l+ohNLmBqFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8e4f6432e67ebf6a0132e515563fccd152dc40c1321436cb1b6509ae4b6d7b0","last_reissued_at":"2026-05-18T02:47:43.067844Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:43.067844Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Birth and death process with one-side bounded jumps in random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hua-Ming Wang","submitted_at":"2014-07-12T13:42:41Z","abstract_excerpt":"Let $\\omega=(\\omega_i)_{i\\in\\mathbb Z}=(\\mu^{L}_i,...,\\mu^{1}_i,\\lambda_i)_{i\\in \\mathbb Z}$, which serves as the environment, be a sequence of i.i.d. random nonnegative vectors, with $L\\ge1$ a positive integer. We study birth and death process $N_t$ which, given the environment $\\omega,$ waits at a state $n$ an exponentially distributed time with parameter $\\lambda_n+\\sum_{l=1}^L\\mu^{l}_n$ and then jumps to $n-i$ with probability ${\\mu^i_n}/(\\lambda_n+\\sum_{l=1}^L\\mu^{l}_n),$ $i=1,...,L$ or to $n+1$ with probability ${\\lambda_n}/(\\lambda_n+\\sum_{l=1}^L\\mu^{l}_n).$ A sufficient condition for t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.3385","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1407.3385","created_at":"2026-05-18T02:47:43.067973+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.3385v1","created_at":"2026-05-18T02:47:43.067973+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.3385","created_at":"2026-05-18T02:47:43.067973+00:00"},{"alias_kind":"pith_short_12","alias_value":"3DSPMQZOM7V7","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_16","alias_value":"3DSPMQZOM7V7NIAT","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_8","alias_value":"3DSPMQZO","created_at":"2026-05-18T12:28:11.866339+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3DSPMQZOM7V7NIATFZIVKY74ZU","json":"https://pith.science/pith/3DSPMQZOM7V7NIATFZIVKY74ZU.json","graph_json":"https://pith.science/api/pith-number/3DSPMQZOM7V7NIATFZIVKY74ZU/graph.json","events_json":"https://pith.science/api/pith-number/3DSPMQZOM7V7NIATFZIVKY74ZU/events.json","paper":"https://pith.science/paper/3DSPMQZO"},"agent_actions":{"view_html":"https://pith.science/pith/3DSPMQZOM7V7NIATFZIVKY74ZU","download_json":"https://pith.science/pith/3DSPMQZOM7V7NIATFZIVKY74ZU.json","view_paper":"https://pith.science/paper/3DSPMQZO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.3385&json=true","fetch_graph":"https://pith.science/api/pith-number/3DSPMQZOM7V7NIATFZIVKY74ZU/graph.json","fetch_events":"https://pith.science/api/pith-number/3DSPMQZOM7V7NIATFZIVKY74ZU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3DSPMQZOM7V7NIATFZIVKY74ZU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3DSPMQZOM7V7NIATFZIVKY74ZU/action/storage_attestation","attest_author":"https://pith.science/pith/3DSPMQZOM7V7NIATFZIVKY74ZU/action/author_attestation","sign_citation":"https://pith.science/pith/3DSPMQZOM7V7NIATFZIVKY74ZU/action/citation_signature","submit_replication":"https://pith.science/pith/3DSPMQZOM7V7NIATFZIVKY74ZU/action/replication_record"}},"created_at":"2026-05-18T02:47:43.067973+00:00","updated_at":"2026-05-18T02:47:43.067973+00:00"}