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It is known that if $\\mathbb{E}(\\log^{+}\\|X_1\\|)< \\infty$, then with probability one for all $1 \\leq p \\leq d$, $$\n  \\lim_{n \\to \\infty} \\frac{1}{n}\\log \\sigma_p(n)=\\gamma_p, $$ where ${\\gamma_1,\\gamma_2 \\ldots \\gamma_d}$ are 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":"1606.07704","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-24T14:35:22Z","cross_cats_sorted":["math-ph","math.CA","math.MP"],"title_canon_sha256":"94d40376d6c6d19c340f88382013ba11a80a1b1e5290cdbcf1336457de032afd","abstract_canon_sha256":"a5956a20f09f87442861df50822974baf8178cf7e48c7ac1b8dfa1a70f506395"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:55.863922Z","signature_b64":"4z58g99n99eiQwtONp1eh8LdrlgFF2yIqi5qivSPYFRGF3JWgL1rTUxrabJWKvrQnWvAOJ5GBs+Bznery2imBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f35701471965fa82d0eac2a07c8c764e7ec485f97c333253a98468471710df87","last_reissued_at":"2026-05-18T01:11:55.863589Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:55.863589Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lyapunov exponents and eigenvalues of products of random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CA","math.MP"],"primary_cat":"math.PR","authors_text":"Nanda Kishore Reddy","submitted_at":"2016-06-24T14:35:22Z","abstract_excerpt":"Let $X_1,X_2, \\ldots $ be a sequence of $i.i.d$ real (complex) $d \\times d $ invertible random matrices with common distribution $\\mu$ and $\\sigma_1(n), \\sigma_2(n), \\ldots , \\sigma_d(n)$ be the singular values, $\\lambda_1(n), \\lambda_2(n), \\ldots , \\lambda_d(n)$ be the eigenvalues of $X_nX_{n-1}\\cdots X_1$ in the decreasing order of their absolute values for every $n$. It is known that if $\\mathbb{E}(\\log^{+}\\|X_1\\|)< \\infty$, then with probability one for all $1 \\leq p \\leq d$, $$\n  \\lim_{n \\to \\infty} \\frac{1}{n}\\log \\sigma_p(n)=\\gamma_p, $$ where ${\\gamma_1,\\gamma_2 \\ldots \\gamma_d}$ are t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07704","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":"1606.07704","created_at":"2026-05-18T01:11:55.863645+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.07704v1","created_at":"2026-05-18T01:11:55.863645+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.07704","created_at":"2026-05-18T01:11:55.863645+00:00"},{"alias_kind":"pith_short_12","alias_value":"6NLQCRYZMX5I","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_16","alias_value":"6NLQCRYZMX5IFUHK","created_at":"2026-05-18T12:30:01.593930+00:00"},{"alias_kind":"pith_short_8","alias_value":"6NLQCRYZ","created_at":"2026-05-18T12:30:01.593930+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/6NLQCRYZMX5IFUHKYKQHZDDWJZ","json":"https://pith.science/pith/6NLQCRYZMX5IFUHKYKQHZDDWJZ.json","graph_json":"https://pith.science/api/pith-number/6NLQCRYZMX5IFUHKYKQHZDDWJZ/graph.json","events_json":"https://pith.science/api/pith-number/6NLQCRYZMX5IFUHKYKQHZDDWJZ/events.json","paper":"https://pith.science/paper/6NLQCRYZ"},"agent_actions":{"view_html":"https://pith.science/pith/6NLQCRYZMX5IFUHKYKQHZDDWJZ","download_json":"https://pith.science/pith/6NLQCRYZMX5IFUHKYKQHZDDWJZ.json","view_paper":"https://pith.science/paper/6NLQCRYZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.07704&json=true","fetch_graph":"https://pith.science/api/pith-number/6NLQCRYZMX5IFUHKYKQHZDDWJZ/graph.json","fetch_events":"https://pith.science/api/pith-number/6NLQCRYZMX5IFUHKYKQHZDDWJZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6NLQCRYZMX5IFUHKYKQHZDDWJZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6NLQCRYZMX5IFUHKYKQHZDDWJZ/action/storage_attestation","attest_author":"https://pith.science/pith/6NLQCRYZMX5IFUHKYKQHZDDWJZ/action/author_attestation","sign_citation":"https://pith.science/pith/6NLQCRYZMX5IFUHKYKQHZDDWJZ/action/citation_signature","submit_replication":"https://pith.science/pith/6NLQCRYZMX5IFUHKYKQHZDDWJZ/action/replication_record"}},"created_at":"2026-05-18T01:11:55.863645+00:00","updated_at":"2026-05-18T01:11:55.863645+00:00"}