Agafonov's Theorem for finite and infinite alphabets and probability distributions different from equidistribution

Abstract

An infinite sequence $\alpha$ over an alphabet $\Sigma$ is $\mu$-distributed w.r.t. a probability map $\mu$ if, for every finite string $w$, the limiting frequency of $w$ in $\alpha$ exists and equals $\mu(w)$. %We raise the question of how to characterize the probability maps $\mu$ for which $\mu$-distributedness is preserved across finite-state selection, or equivalently, by selection by programs using constant space. We prove the following result for any finite or countably infinite alphabet $\Sigma$: every finite-state selector over $\Sigma$ selects a $\mu$-distributed sequence from every $\mu$-distributed sequence \emph{if and only if} $\mu$ is induced by a Bernoulli distribution on $\Sigma$, that is a probability distribution on the alphabet extended to words by taking the product. The primary -- and remarkable -- consequence of our main result is a complete characterization of the set of probability maps, on finite and infinite alphabets, for which finite-state selection preserves $\mu$-distributedness. The main positive takeaway is that (the appropriate generalization of) Agafonov's Theorem holds for Bernoulli distributions (rather than just equidistributions) on both finite and countably infinite alphabets. As a further consequence, we obtain a result in the area of symbolic dynamical systems: the shift-invariant measures $\mu$ on $\Sigma^{\omega}$ such that any finite-state selector preserves the property of genericity for $\mu$, are exactly the positive Bernoulli measures.