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

Abstract

An infinite sequence over a finite alphabet of symbols Σ is called normal iff the limiting frequency of every finite string w ∈ Σ * exists and equals |Σ| −|w| .A celebrated theorem by Agafonov states that a sequence α is normal iff every finitestate selector (i.e., a DFA accepting or rejecting prefixes of α) selects a normal sequence from α.Let µ : Σ * −→ [0, 1] be a probability map (for every n ≥ 0, w∈Σ n µ(w) = 1). Say that an infinite sequence α is is µ-distributed if, for every w ∈ Σ * , the limiting frequency of w in α exists and equals µ(w). Thus, α is normal if it is µ-distributed for the probability map µ(w) = |Σ| −|w| .Unlike normality, µ-distributedness is not preserved by finite-state selectors for all probability maps µ. This raises the question of how to characterize the probability maps µ for which µ-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 Σ, every finite-state selector over Σ selects a µ-distributed sequence from every µ-distributed sequence α iff µ is induced by a Bernoulli distribution on Σ, that is, for every wordThe 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 ν on Σ ω such that any finite-state selector preserves the property of genericity for ν are exactly the positive Bernoulli measures.