Summary
We show that almost all binary strings of length n contain all blocks of size (1-ε)log2 n a close to uniform number of times. From this, we derive tight bounds on the discrepancy of random infinite strings. Our results are obtained through explicit generating function expressions and contour integration estimates.
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Research of the three authors was supported by the French-Austrian scientific cooperation programme
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Flajolet, P., Kirschenhofer, P. & Tichy, R.F. Deviations from uniformity in random strings. Probab. Th. Rel. Fields 80, 139–150 (1988). https://doi.org/10.1007/BF00348756
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DOI: https://doi.org/10.1007/BF00348756