Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Agafonov's Theorem for finite and infinite alphabets and probability
  distributions different from equidistribution   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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The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga