Finite-state dimension quantifies the asymptotic rate of information in an infinite sequence as perceived by finite automata. For a fixed alphabet, the infinite sequences that have maximal finite-state dimension are exactly those that are Borel normal, i.e., in which all words of any given length appear with the same frequency. A theorem of Schnorr and Stimm (1972) shows that a real number is Borel normal if and only if, for every finite-state irreducible Markov chain with fair transitions, when the chain is simulated using the binary expansion of the given number, the empirical distribution of states converges to its stationary distribution. In this paper we extend this correspondence beyond normal numbers. We show that the finite-state dimension of a sequence can be characterized in terms of the conditional Kullback-Leibler divergence between the limiting distributions arising from the simulation of Markov chains using the given sequence and their stationary distributions. This provides a new information-theoretic characterization of finite-state dimension which generalizes the Schnorr-Stimm result. As an application, we prove a generalization of Agafonov's theorem for normal numbers. Agafonov's theorem states that a sequence is normal if and only if every subsequence selected by a finite automaton is also normal. We extend this to arbitrary sequences by establishing a tight quantitative relationship between the finite-state dimension of a sequence and the finite-state dimensions of its automatic subsequences.

翻译：有限状态维度量化了有限自动机所感知的无限序列中信息的渐近速率。对于固定字母表，具有最大有限状态维度的无限序列恰好是那些Borel正规序列，即在其中任何给定长度的所有词都以相同频率出现。Schnorr与Stimm（1972）的定理表明，当且仅当对于每个具有公平转移的有限状态不可约马尔可夫链，使用给定数字的二进制展开模拟该链时，状态的经验分布收敛于其平稳分布，该实数才是Borel正规的。本文将此对应关系推广至正规数之外。我们证明序列的有限状态维度可以通过使用给定序列模拟马尔可夫链产生的极限分布与其平稳分布之间的条件Kullback-Leibler散度来刻画。这为有限状态维度提供了新的信息论特征，推广了Schnorr-Stimm的结果。作为应用，我们证明了正规数的阿加福诺夫定理的推广。阿加福诺夫定理指出，序列是正规的当且仅当由有限自动机选择的每个子序列也是正规的。我们通过建立序列的有限状态维度与其自动子序列的有限状态维度之间的严格定量关系，将此结果推广至任意序列。