{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:D7QZ6ND3T5NZRXCXWGDZMDPSAO","short_pith_number":"pith:D7QZ6ND3","schema_version":"1.0","canonical_sha256":"1fe19f347b9f5b98dc57b187960df203be03a26103a051e95e5ec01d3a7d82de","source":{"kind":"arxiv","id":"1610.02861","version":3},"attestation_state":"computed","paper":{"title":"A multidimensional analogue of the arcsine law for the number of positive terms in a random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, Vladislav Vysotsky, Zakhar Kabluchko","submitted_at":"2016-10-10T11:46:17Z","abstract_excerpt":"Consider a random walk $S_i= \\xi_1+\\ldots+\\xi_i$, $i\\in\\mathbb N$, whose increments $\\xi_1,\\xi_2,\\ldots$ are independent identically distributed random vectors in $\\mathbb R^d$ such that $\\xi_1$ has the same law as $-\\xi_1$ and $\\mathbb P[\\xi_1\\in H] = 0$ for every affine hyperplane $H\\subset \\mathbb R^d$. Our main result is the distribution-free formula $$ \\mathbb E \\left[\\sum_{1\\leq i_1 < \\ldots < i_k\\leq n} 1_{\\{0\\notin \\text{conv}(S_{i_1},\\ldots, S_{i_k})\\}}\\right] = 2 \\binom n k \\frac {B(k, d-1) + B(k, d-3) +\\ldots} {2^k k!}, $$ where the $B(k,j)$'s are defined by their generating functio"},"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":"1610.02861","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-10-10T11:46:17Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"404fae2c21d9bd3461b68b8b8e35509a42767c2dc08b521f193717f8ab497dd9","abstract_canon_sha256":"7a68e9cbdbcdfb56cc3f0316601d4fe7a3d7c993db2aedc4efc03139cf6e2773"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:37:48.255872Z","signature_b64":"9BhAyOInKJRTMgwFSosVhzBiznDFmW9xegGS34e82Lu+7q8atxV/l+xYkX46vwKv4hHkjbOiGM2eFCYIW9afAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1fe19f347b9f5b98dc57b187960df203be03a26103a051e95e5ec01d3a7d82de","last_reissued_at":"2026-05-18T00:37:48.255161Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:37:48.255161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A multidimensional analogue of the arcsine law for the number of positive terms in a random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, Vladislav Vysotsky, Zakhar Kabluchko","submitted_at":"2016-10-10T11:46:17Z","abstract_excerpt":"Consider a random walk $S_i= \\xi_1+\\ldots+\\xi_i$, $i\\in\\mathbb N$, whose increments $\\xi_1,\\xi_2,\\ldots$ are independent identically distributed random vectors in $\\mathbb R^d$ such that $\\xi_1$ has the same law as $-\\xi_1$ and $\\mathbb P[\\xi_1\\in H] = 0$ for every affine hyperplane $H\\subset \\mathbb R^d$. Our main result is the distribution-free formula $$ \\mathbb E \\left[\\sum_{1\\leq i_1 < \\ldots < i_k\\leq n} 1_{\\{0\\notin \\text{conv}(S_{i_1},\\ldots, S_{i_k})\\}}\\right] = 2 \\binom n k \\frac {B(k, d-1) + B(k, d-3) +\\ldots} {2^k k!}, $$ where the $B(k,j)$'s are defined by their generating functio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02861","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":"1610.02861","created_at":"2026-05-18T00:37:48.255275+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.02861v3","created_at":"2026-05-18T00:37:48.255275+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.02861","created_at":"2026-05-18T00:37:48.255275+00:00"},{"alias_kind":"pith_short_12","alias_value":"D7QZ6ND3T5NZ","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"D7QZ6ND3T5NZRXCX","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"D7QZ6ND3","created_at":"2026-05-18T12:30:09.641336+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/D7QZ6ND3T5NZRXCXWGDZMDPSAO","json":"https://pith.science/pith/D7QZ6ND3T5NZRXCXWGDZMDPSAO.json","graph_json":"https://pith.science/api/pith-number/D7QZ6ND3T5NZRXCXWGDZMDPSAO/graph.json","events_json":"https://pith.science/api/pith-number/D7QZ6ND3T5NZRXCXWGDZMDPSAO/events.json","paper":"https://pith.science/paper/D7QZ6ND3"},"agent_actions":{"view_html":"https://pith.science/pith/D7QZ6ND3T5NZRXCXWGDZMDPSAO","download_json":"https://pith.science/pith/D7QZ6ND3T5NZRXCXWGDZMDPSAO.json","view_paper":"https://pith.science/paper/D7QZ6ND3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.02861&json=true","fetch_graph":"https://pith.science/api/pith-number/D7QZ6ND3T5NZRXCXWGDZMDPSAO/graph.json","fetch_events":"https://pith.science/api/pith-number/D7QZ6ND3T5NZRXCXWGDZMDPSAO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D7QZ6ND3T5NZRXCXWGDZMDPSAO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D7QZ6ND3T5NZRXCXWGDZMDPSAO/action/storage_attestation","attest_author":"https://pith.science/pith/D7QZ6ND3T5NZRXCXWGDZMDPSAO/action/author_attestation","sign_citation":"https://pith.science/pith/D7QZ6ND3T5NZRXCXWGDZMDPSAO/action/citation_signature","submit_replication":"https://pith.science/pith/D7QZ6ND3T5NZRXCXWGDZMDPSAO/action/replication_record"}},"created_at":"2026-05-18T00:37:48.255275+00:00","updated_at":"2026-05-18T00:37:48.255275+00:00"}