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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
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[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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