Mathematical metaphysics of randomness
Theoretical Computer Science - Special issue Kolmogorov complexity
On the Role of the Law of Large Numbers in the Theory of Randomness
Problems of Information Transmission
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We improve a well-known asymptotic bound on the number of monotonic selection rules for covering of an arbitrary randomness test by frequency tests. More precisely, we prove that, for any set S (arbitrary test) of binary sequences of sufficiently large length L, where 驴S驴 驴 2 L(1驴驴), for sufficiently small 驴 there exists a polynomial (in 1/驴) set of monotonic selection rules (frequency tests) which guarantee that, for each sequence t 驴 S, a subsequence can be selected such that the product of its length by the squared deviation of the fraction of zeros in it from 1/2 is of the order of at least 0.5 ln 2 L[驴/ln(1/驴)](1 驴 2 ln ln(1/驴)/ln(1/驴)).