混合序列Kolmogorov复杂度的高斯性
On the Gaussianity of Kolmogorov Complexity of Mixing Sequences
摘要：
令$ K(X_1, \ldots, X_n)$和$H(X_n | X_{n-1}, \ldots, X_1)$表示平稳遍历过程$\{X_i\}_{i=-\infty}^\infty$的Kolmogorov复杂度和Shannon熵率。已证明\[ \frac{K(X_1, \ldots, X_n)}{n} - H(X_n | X_{n-1}, \ldots, X_1) \rightarrow 0, \]几乎肯定。本文研究了这一渐近结果的收敛速度。特别是，我们表明，如果该过程满足一定的混合条件，则存在$σ<\infty$，使$$\sqrt{n}\left(\frac{K(X_{1:n})}{n}- H(X_0|X_1,\dots,X_{-\infty})\right) \rightarrow_d N(0,σ^2).$$此外，我们表明，在稍强的混合条件下，可以获得非渐近浓度边界的柯尔莫哥洛夫复杂性。
英文摘要：
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 $σ<\infty$ such that $$\sqrt{n}\left(\frac{K(X_{1:n})}{n}- H(X_0|X_1,\dots,X_{-\infty})\right) \rightarrow_d N(0,σ^2).$$ Furthermore, we show that under slightly stronger mixing conditions one may obtain non-asymptotic concentration bounds for the Kolmogorov complexity.
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