{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2005:2TP67LXW3SFKVGVPWHT5C7II4B","short_pith_number":"pith:2TP67LXW","schema_version":"1.0","canonical_sha256":"d4dfefaef6dc8aaa9aafb1e7d17d08e06c2f6642cb3b2a90dc0e7afea21ea3e8","source":{"kind":"arxiv","id":"math/0508371","version":3},"attestation_state":"computed","paper":{"title":"Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"Alexandra Rodkina, Gregory Berkolaiko","submitted_at":"2005-08-19T16:35:03Z","abstract_excerpt":"We consider a non-homogeneous nonlinear stochastic difference equation\n  X_{n+1} = X_n (1 + f(X_n)\\xi_{n+1}) + S_n, and its important special case\n  X_{n+1} = X_n (1 + \\xi_{n+1}) + S_n, both with initial value X_0, non-random decaying free coefficient S_n and independent random variables \\xi_n. We establish results on \\as convergence of solutions X_n to zero. The necessary conditions we find tie together certain moments of the noise \\xi_n and the rate of decay of S_n. To ascertain sharpness of our conditions we discuss some situations when X_n diverges. We also establish a result concerning th"},"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/0508371","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2005-08-19T16:35:03Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"bebec9ac898d22624e75ea7d6544dab8fc1aa330beb3456327e8db54f8bd9ac9","abstract_canon_sha256":"c3e479465bb4981db82991b054cf759a8d254e925f2f94b4b6d55da5acc82ace"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:47.336128Z","signature_b64":"uhPqQr0Xr76t5SRNNOf+dDwLwwNz8HHSPrCPGuPEJTplaJFr8jQdhnx+xTSTJRE3iKIHiLOCIurXYPa+aDFaAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4dfefaef6dc8aaa9aafb1e7d17d08e06c2f6642cb3b2a90dc0e7afea21ea3e8","last_reissued_at":"2026-05-18T04:10:47.335722Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:47.335722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"Alexandra Rodkina, Gregory Berkolaiko","submitted_at":"2005-08-19T16:35:03Z","abstract_excerpt":"We consider a non-homogeneous nonlinear stochastic difference equation\n  X_{n+1} = X_n (1 + f(X_n)\\xi_{n+1}) + S_n, and its important special case\n  X_{n+1} = X_n (1 + \\xi_{n+1}) + S_n, both with initial value X_0, non-random decaying free coefficient S_n and independent random variables \\xi_n. We establish results on \\as convergence of solutions X_n to zero. The necessary conditions we find tie together certain moments of the noise \\xi_n and the rate of decay of S_n. To ascertain sharpness of our conditions we discuss some situations when X_n diverges. We also establish a result concerning th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508371","kind":"arxiv","version":3},"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":"math/0508371","created_at":"2026-05-18T04:10:47.335779+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0508371v3","created_at":"2026-05-18T04:10:47.335779+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0508371","created_at":"2026-05-18T04:10:47.335779+00:00"},{"alias_kind":"pith_short_12","alias_value":"2TP67LXW3SFK","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_16","alias_value":"2TP67LXW3SFKVGVP","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_8","alias_value":"2TP67LXW","created_at":"2026-05-18T12:25:52.687210+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/2TP67LXW3SFKVGVPWHT5C7II4B","json":"https://pith.science/pith/2TP67LXW3SFKVGVPWHT5C7II4B.json","graph_json":"https://pith.science/api/pith-number/2TP67LXW3SFKVGVPWHT5C7II4B/graph.json","events_json":"https://pith.science/api/pith-number/2TP67LXW3SFKVGVPWHT5C7II4B/events.json","paper":"https://pith.science/paper/2TP67LXW"},"agent_actions":{"view_html":"https://pith.science/pith/2TP67LXW3SFKVGVPWHT5C7II4B","download_json":"https://pith.science/pith/2TP67LXW3SFKVGVPWHT5C7II4B.json","view_paper":"https://pith.science/paper/2TP67LXW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0508371&json=true","fetch_graph":"https://pith.science/api/pith-number/2TP67LXW3SFKVGVPWHT5C7II4B/graph.json","fetch_events":"https://pith.science/api/pith-number/2TP67LXW3SFKVGVPWHT5C7II4B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2TP67LXW3SFKVGVPWHT5C7II4B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2TP67LXW3SFKVGVPWHT5C7II4B/action/storage_attestation","attest_author":"https://pith.science/pith/2TP67LXW3SFKVGVPWHT5C7II4B/action/author_attestation","sign_citation":"https://pith.science/pith/2TP67LXW3SFKVGVPWHT5C7II4B/action/citation_signature","submit_replication":"https://pith.science/pith/2TP67LXW3SFKVGVPWHT5C7II4B/action/replication_record"}},"created_at":"2026-05-18T04:10:47.335779+00:00","updated_at":"2026-05-18T04:10:47.335779+00:00"}