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doi:10.1016/j.ipl.2004.05.014 
Copyright © 2004 Elsevier B.V. All rights reserved.
Kolmogorov–Loveland stochasticity for finite strings
Received 18 June 2002;
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Abstract
Asarin [Theory Probab. Appl. 32 (1987) 507–508] showed that any finite sequence with small randomness deficiency has the stability property of the frequency of 1s in their subsequences selected by simple Kolmogorov–Loveland selection rules. Roughly speaking the difference between frequency m/n of zeros and 1/2 in a subsequence of length n selected from a sequence with randomness deficiency d by a selection rule of complexity k is bounded by
in absolute value. In this paper we prove a result in the inverse direction: if the randomness deficiency of a sequence is large then there is a simple Kolmogorov–Loveland selection rule that selects not too short subsequence in which frequency of ones is far from 1/2. Roughly speaking for any sequence of length N there is a selection rule of complexity O(log(N/d)) selecting a subsequence such that |m/n−1/2|=Ω(d/(nlog(N/d))).
Author Keywords: Kolmogorov–Loveland selection; Finite string; Algorithms
