{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:6Q6K64R2ESSLHHSGT5B7MQBVEU","short_pith_number":"pith:6Q6K64R2","schema_version":"1.0","canonical_sha256":"f43caf723a24a4b39e469f43f64035250ab653b84290453f885f93749bbd3886","source":{"kind":"arxiv","id":"1508.06550","version":1},"attestation_state":"computed","paper":{"title":"Asymptotics for randomly reinforced urns with random barriers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Irene Crimaldi, Luca Pratelli, Patrizia Berti, Pietro Rigo","submitted_at":"2015-08-26T16:14:34Z","abstract_excerpt":"An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\\leq L<U\\leq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together with a random number $B_n$ of black balls. If $b_n$ is red and $Z_{n-1}>L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_n\\overset{a.s.}\\longrightarrow Z$ for some random variab"},"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":"1508.06550","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-26T16:14:34Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"193da96ec20a3f04573c99a79f0ff108d0d24e45cc04443eaa7a76a96fb0f1e0","abstract_canon_sha256":"efe92aa1a9fb1dbed48d0322213fbeec796e40e7f7946b6fc53d70ef4ef8cfa2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:42.729691Z","signature_b64":"k2LnZdEUHiJMCb+zi4mKHV7djzQKSbjCdlr6lLU1FpouxOHvg0bjQr3UzP8rr1FSfRKwOQ0lXvdCI5tIyjyXBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f43caf723a24a4b39e469f43f64035250ab653b84290453f885f93749bbd3886","last_reissued_at":"2026-05-18T01:34:42.729142Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:42.729142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics for randomly reinforced urns with random barriers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Irene Crimaldi, Luca Pratelli, Patrizia Berti, Pietro Rigo","submitted_at":"2015-08-26T16:14:34Z","abstract_excerpt":"An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\\leq L<U\\leq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together with a random number $B_n$ of black balls. If $b_n$ is red and $Z_{n-1}>L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_n\\overset{a.s.}\\longrightarrow Z$ for some random variab"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06550","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":"1508.06550","created_at":"2026-05-18T01:34:42.729214+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.06550v1","created_at":"2026-05-18T01:34:42.729214+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06550","created_at":"2026-05-18T01:34:42.729214+00:00"},{"alias_kind":"pith_short_12","alias_value":"6Q6K64R2ESSL","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_16","alias_value":"6Q6K64R2ESSLHHSG","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_8","alias_value":"6Q6K64R2","created_at":"2026-05-18T12:29:07.941421+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/6Q6K64R2ESSLHHSGT5B7MQBVEU","json":"https://pith.science/pith/6Q6K64R2ESSLHHSGT5B7MQBVEU.json","graph_json":"https://pith.science/api/pith-number/6Q6K64R2ESSLHHSGT5B7MQBVEU/graph.json","events_json":"https://pith.science/api/pith-number/6Q6K64R2ESSLHHSGT5B7MQBVEU/events.json","paper":"https://pith.science/paper/6Q6K64R2"},"agent_actions":{"view_html":"https://pith.science/pith/6Q6K64R2ESSLHHSGT5B7MQBVEU","download_json":"https://pith.science/pith/6Q6K64R2ESSLHHSGT5B7MQBVEU.json","view_paper":"https://pith.science/paper/6Q6K64R2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.06550&json=true","fetch_graph":"https://pith.science/api/pith-number/6Q6K64R2ESSLHHSGT5B7MQBVEU/graph.json","fetch_events":"https://pith.science/api/pith-number/6Q6K64R2ESSLHHSGT5B7MQBVEU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6Q6K64R2ESSLHHSGT5B7MQBVEU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6Q6K64R2ESSLHHSGT5B7MQBVEU/action/storage_attestation","attest_author":"https://pith.science/pith/6Q6K64R2ESSLHHSGT5B7MQBVEU/action/author_attestation","sign_citation":"https://pith.science/pith/6Q6K64R2ESSLHHSGT5B7MQBVEU/action/citation_signature","submit_replication":"https://pith.science/pith/6Q6K64R2ESSLHHSGT5B7MQBVEU/action/replication_record"}},"created_at":"2026-05-18T01:34:42.729214+00:00","updated_at":"2026-05-18T01:34:42.729214+00:00"}