On the Gaussianity of Kolmogorov Complexity of Mixing Sequences

Let KpX1, . . . , Xnq and HpXn|Xn ́1, . . . , X1q denote the Kolmogorov complexity and Shannon’s entropy rate of a stationary and ergodic process tXiui“ ́8. It has been proved that KpX1, . . . , Xnq n ́HpXn|Xn ́1, . . . , X1q Ñ 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 ă 8 such that ? n ˆ KpX1:nq n ́HpX0|X1, . . . , X ́8q ̇ Ñd Np0, σq. Furthermore, we show that under slightly stronger mixing conditions one may obtain nonasymptotic concentration bounds for the Kolmogorov complexity.