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A note on randomness. (English) Zbl 0809.60038
Summary: We give a characterization of how fast the block length $$k$$ may grow in order to have the relative frequence of occurrences of blocks of length $$k$$ among the first $$n$$ tosses of a fair coin be ‘good’ approximations of their probabilities. The borderline will roughly be the logarithm of $$n$$ with base 2, with the details depending on the definition of ‘good’. This extends results by P. Flajolet, P. Kirschenhofer and R. F. Tichy [Probab. Theory Relat. Fields 80, No. 1, 139-150 (1988; Zbl 0638.68058)].

##### MSC:
 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks 60F10 Large deviations
##### Keywords:
coin tossing; uniform distribution
Full Text:
##### References:
 [1] Flajolet, P.; Kirschenhofer, P.; Tichy, R., Deviations from normality in random strings, Probab. theory rel. fields, 80, 139-150, (1988) · Zbl 0638.68058 [2] Guibas, L.; Odlyzko, A., Long repetitive patterns in random sequences, Z. wahrsch. verw. gebiete, 53, 241-262, (1980) · Zbl 0424.60036 [3] Hlawka, E., Theorie der gleichverteilung, (1979), Bibl. Inst Mannheim · Zbl 0406.10001 [4] Knuth, D.E., The art of computer programming, 2, (1969), Addison-Wesley Reading, MA · Zbl 0191.17903 [5] Kuipers, L.; Niederreiter, H., Uniform distribution of sequences, (1974), Wiley New York · Zbl 0281.10001
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