On the Connection between the Concepts of Collectives of Mises-Church and Normal Bernoulli Sequences of Symbols

Постникова, Л. П.

doi:10.1137/1106028

Cited by 5 publications

(2 citation statements)

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“…Both Postnikova [52] and Agafonov [45] considered selection functions on sequences in {0, 1} ω where the limiting distribution of 1 was 0 < p < 1 (i.e., considered a Bernoulli distribution on {0, 1}), but considering Bernoulli distributions instead of the special case of equidistributions seems to have disappeared almost completely from all later work 2 . For equidistribution, the earliest extension to arbitrary alphabets seems to be by Broglio and Liardet [15], and a number of authors have since re-proved Agafonov's Theorem in the special case of equidistribution using a variety of methods; for example, using predictors defined from finite automata (for Σ = {0, 1}) [44], using compressibility arguments [6,5,58], and a combination of automata-theoretic and probabilistic methods similar to Agafonov's original reasoning [16].…”

Section: Agafonov's Theorem and Its Generalizationsmentioning

confidence: 99%

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

Seiller¹,

Simonsen²

2020

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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.

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Section: Agafonov's Theorem and Its Generalizationsmentioning

confidence: 99%

“…The exact definition of kollektiv differs subtly across different authors, compare e.g [67],[21],. and[52]. The original notion of kollektiv introduced by von Mises[67] had no constraints on the set S, but this turned out to be essentially fruitless[63,53,32,23].…”

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confidence: 99%