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

Abstract : An infinite sequence over a finite alphabet {\Sigma} of symbols is called normal iff the limiting frequency of every finite string w exists and equals |{\Sigma}|^{|w|}. A celebrated theorem by Agafonov states that a sequence {\alpha} is normal iff every finite-state selector. Normality is generalised to arbitrary probability maps \mu: {\alpha} is is \mu-distributed if, for every finite string w, the limiting frequency of w in {\alpha} exists and equals \mu(w). Unlike normality, \mu-distributedness is not preserved by finite-state selectors for all probability maps \mu. This raises 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 {\alpha} iff \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 Agafonov-type results hold. 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 {\nu} on {\Sigma}^{\omega} such that any finite-state selector preserves the property of genericity for {\mu}, are exactly the positive Bernoulli measures.
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https://hal.archives-ouvertes.fr/hal-02993635
Contributor : Thomas Seiller <>
Submitted on : Friday, November 6, 2020 - 9:05:36 PM
Last modification on : Wednesday, November 18, 2020 - 8:28:13 AM

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• HAL Id : hal-02993635, version 1
• ARXIV : 2011.08552

### Citation

Thomas Seiller, Jakob Simonsen. Agafonov's Theorem for finite and infinite alphabets and probability distributions different from equidistribution. 2020. ⟨hal-02993635⟩

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