On the Gaussianity of Kolmogorov Complexity of Mixing Sequences

Abstract

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 $$\sqrt{n}\left(\frac{K(X_{1:n})}{n}- H(X_0|X_1,\dots,X_{-\infty})\right) \rightarrow_d N(0,\sigma^2).$$ Furthermore, we show that under slightly stronger mixing conditions one may obtain non-asymptotic concentration bounds for the Kolmogorov complexity.