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Let $A^{\\el}$ denote $A_1^{\\ell_1} ...                                                                                                                                                                            \nA_d^{\\ell_d}$, for ${\\el}= (\\ell_1, ..., \\ell_d)$. If $(Z_k)$ is a random walk on $\\Z^d$, one can study the asymptotic distribution of the sums $\\sum_{k=0}^{n-1} \\, f \\circ A^{\\,{Z_k(\\omega)}}$\nand $\\sum_{\\el \\in \\Z^d} \\PP(Z_n= \\el) \\, A^\\el"},"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":"1411.3540","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-11-13T13:39:00Z","cross_cats_sorted":[],"title_canon_sha256":"c94feb1b421c9b1d5058ca391554bf4053b73671f82f2c5ea5def3465cf47eef","abstract_canon_sha256":"4580005cf8c38c1a01176c03ba0fe43b221063a8b3cedeb3d4399bd9af182b9a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:37:43.500309Z","signature_b64":"//Z53IR8aIuzTOwGv80/4U0+AL8z04JV+sPU/xBgegWhkY0lj+Wcy56trDAusEDCszq0evzHuo5ylAFpwu9RCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"217dd9ce02eb5d979df889ff9c990326c2868931fa7d873000ccb0bcf8d28714","last_reissued_at":"2026-05-18T02:37:43.499912Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:37:43.499912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"CLT for random walks of commuting endomorphisms on compact abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guy Cohen, Jean-Pierre Conze","submitted_at":"2014-11-13T13:39:00Z","abstract_excerpt":"Let $\\Cal S$ be an abelian group of automorphisms of a probability space $(X, {\\Cal A}, \\mu)$ with a finite system of generators $(A_1, ..., A_d)$. Let $A^{\\el}$ denote $A_1^{\\ell_1} ...                                                                                                                                                                            \nA_d^{\\ell_d}$, for ${\\el}= (\\ell_1, ..., \\ell_d)$. If $(Z_k)$ is a random walk on $\\Z^d$, one can study the asymptotic distribution of the sums $\\sum_{k=0}^{n-1} \\, f \\circ A^{\\,{Z_k(\\omega)}}$\nand $\\sum_{\\el \\in \\Z^d} \\PP(Z_n= \\el) \\, A^\\el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.3540","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":"1411.3540","created_at":"2026-05-18T02:37:43.499972+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.3540v1","created_at":"2026-05-18T02:37:43.499972+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.3540","created_at":"2026-05-18T02:37:43.499972+00:00"},{"alias_kind":"pith_short_12","alias_value":"EF65TTQC5NOZ","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_16","alias_value":"EF65TTQC5NOZPHPY","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_8","alias_value":"EF65TTQC","created_at":"2026-05-18T12:28:25.294606+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/EF65TTQC5NOZPHPYRH7ZZGIDE3","json":"https://pith.science/pith/EF65TTQC5NOZPHPYRH7ZZGIDE3.json","graph_json":"https://pith.science/api/pith-number/EF65TTQC5NOZPHPYRH7ZZGIDE3/graph.json","events_json":"https://pith.science/api/pith-number/EF65TTQC5NOZPHPYRH7ZZGIDE3/events.json","paper":"https://pith.science/paper/EF65TTQC"},"agent_actions":{"view_html":"https://pith.science/pith/EF65TTQC5NOZPHPYRH7ZZGIDE3","download_json":"https://pith.science/pith/EF65TTQC5NOZPHPYRH7ZZGIDE3.json","view_paper":"https://pith.science/paper/EF65TTQC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.3540&json=true","fetch_graph":"https://pith.science/api/pith-number/EF65TTQC5NOZPHPYRH7ZZGIDE3/graph.json","fetch_events":"https://pith.science/api/pith-number/EF65TTQC5NOZPHPYRH7ZZGIDE3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EF65TTQC5NOZPHPYRH7ZZGIDE3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EF65TTQC5NOZPHPYRH7ZZGIDE3/action/storage_attestation","attest_author":"https://pith.science/pith/EF65TTQC5NOZPHPYRH7ZZGIDE3/action/author_attestation","sign_citation":"https://pith.science/pith/EF65TTQC5NOZPHPYRH7ZZGIDE3/action/citation_signature","submit_replication":"https://pith.science/pith/EF65TTQC5NOZPHPYRH7ZZGIDE3/action/replication_record"}},"created_at":"2026-05-18T02:37:43.499972+00:00","updated_at":"2026-05-18T02:37:43.499972+00:00"}