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It has been proved that \\[ \\frac{K(X_1, \\ldots, X_n)}{n} - H(X_n | X_{n-1}, \\ldots, X_1) \\rightarrow 0, \\] almost surely. This paper studies the convergence rate of this asymptotic result. In particular, we show that if the process satisfies certain mixing conditions, then there exists $\\sigma<\\infty$ such that $$\\sqrt{n}\\left(\\frac{K(X_{1:n})}{n}- H(X_0|X_1,\\dots,X_{-\\infty})\\right) \\rightarrow_d N(0,\\sigma"},"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":"1702.01317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2017-02-04T18:15:45Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"3e496963f1d0889423333ce1b0dc4d6daf66585176d0af0b19a473bab248c947","abstract_canon_sha256":"0fbf514fc2f1c781c6a3d28b1ad1c642340b06e454b68c9c1feeac9eca68c385"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:23.789542Z","signature_b64":"9rSCWzAz6TSqHvFASG2ytnimpIY3CZF5dZPAYpQGJWwbqAKO8mGDwzEyEWxPUXsPwepugS9XEczS4UHVaU0NAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7f8b65be4d90d577ca4787cce5baf0863180a47230026e5679c7205b0e67d79","last_reissued_at":"2026-05-18T00:51:23.789028Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:23.789028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Gaussianity of Kolmogorov Complexity of Mixing Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Arian Maleki, Morgane Austern","submitted_at":"2017-02-04T18:15:45Z","abstract_excerpt":"Let $ K(X_1, \\ldots, X_n)$ and $H(X_n | X_{n-1}, \\ldots, X_1)$ denote the Kolmogorov complexity and Shannon's entropy rate of a stationary and ergodic process $\\{X_i\\}_{i=-\\infty}^\\infty$. It has been proved that \\[ \\frac{K(X_1, \\ldots, X_n)}{n} - H(X_n | X_{n-1}, \\ldots, X_1) \\rightarrow 0, \\] almost surely. This paper studies the convergence rate of this asymptotic result. In particular, we show that if the process satisfies certain mixing conditions, then there exists $\\sigma<\\infty$ such that $$\\sqrt{n}\\left(\\frac{K(X_{1:n})}{n}- H(X_0|X_1,\\dots,X_{-\\infty})\\right) \\rightarrow_d N(0,\\sigma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01317","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":"1702.01317","created_at":"2026-05-18T00:51:23.789102+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.01317v1","created_at":"2026-05-18T00:51:23.789102+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.01317","created_at":"2026-05-18T00:51:23.789102+00:00"},{"alias_kind":"pith_short_12","alias_value":"474LMW7E3EGV","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"474LMW7E3EGVO7FE","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"474LMW7E","created_at":"2026-05-18T12:30:58.224056+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/474LMW7E3EGVO7FEPB6M4W5PBB","json":"https://pith.science/pith/474LMW7E3EGVO7FEPB6M4W5PBB.json","graph_json":"https://pith.science/api/pith-number/474LMW7E3EGVO7FEPB6M4W5PBB/graph.json","events_json":"https://pith.science/api/pith-number/474LMW7E3EGVO7FEPB6M4W5PBB/events.json","paper":"https://pith.science/paper/474LMW7E"},"agent_actions":{"view_html":"https://pith.science/pith/474LMW7E3EGVO7FEPB6M4W5PBB","download_json":"https://pith.science/pith/474LMW7E3EGVO7FEPB6M4W5PBB.json","view_paper":"https://pith.science/paper/474LMW7E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.01317&json=true","fetch_graph":"https://pith.science/api/pith-number/474LMW7E3EGVO7FEPB6M4W5PBB/graph.json","fetch_events":"https://pith.science/api/pith-number/474LMW7E3EGVO7FEPB6M4W5PBB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/474LMW7E3EGVO7FEPB6M4W5PBB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/474LMW7E3EGVO7FEPB6M4W5PBB/action/storage_attestation","attest_author":"https://pith.science/pith/474LMW7E3EGVO7FEPB6M4W5PBB/action/author_attestation","sign_citation":"https://pith.science/pith/474LMW7E3EGVO7FEPB6M4W5PBB/action/citation_signature","submit_replication":"https://pith.science/pith/474LMW7E3EGVO7FEPB6M4W5PBB/action/replication_record"}},"created_at":"2026-05-18T00:51:23.789102+00:00","updated_at":"2026-05-18T00:51:23.789102+00:00"}