{"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"}