no. 2
Block distribution in random strings
Annales de l'Institut Fourier, Tome 43 (1993) no. 2, pp. 539-549.

Pour presque toute suite binaire infinie issue d’un tirage de Bernoulli (p,q) la fréquence des blocs de longueur k(N) dans les N premiers termes tend asymptotiquement vers la probabilité naturelle du bloc, ceci lorsque k(N) croît comme log 1 p N- log 1 p N-ψ(N) (avec p≤q) où ψ(N) tend vers +∞. Ce résultat généralise celui de P. Flajolet, P. Kirschenhofer et R.F. Tichy concernant le cas uniforme p=q=1 2.

For almost all infinite binary sequences of Bernoulli trials (p,q) the frequency of blocks of length k(N) in the first N terms tends asymptotically to the probability of the blocks, if k(N) increases like log 1 p N- log 1 p N-ψ(N) (for p≤q) where ψ(N) tends to +∞. This generalizes a result due to P. Flajolet, P. Kirschenhofer and R.F. Tichy concerning the case p=q=1 2.

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     author = {Grabner, Peter J.},
     title = {Block distribution in random strings},
     journal = {Annales de l'Institut Fourier},
     pages = {539--549},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {2},
     year = {1993},
     doi = {10.5802/aif.1345},
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     mrnumber = {94d:60044},
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     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1345/}
}
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Grabner, Peter J. Block distribution in random strings. Annales de l'Institut Fourier, Tome 43 (1993) no. 2, pp. 539-549. doi : 10.5802/aif.1345. https://aif.centre-mersenne.org/articles/10.5802/aif.1345/

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[FKT] P. Flajolet, P. Kirschenhofer and R.F. Tichy, Deviations from Uniformity in Random Strings, Probab. Th. Rel. Fields, vol 80 (1988), 139-150. | MR | Zbl

[Gr] K. Grill, A Note on Randomness, Stat. and Probab. Letters, to appear. | Zbl

[GO] L. Guibas and A.M. Odlyzko, String Overlaps, Pattern Matching and Non-transitive Games, J. Comb. Th., Ser A, vol 30 (1981), 183-208. | Zbl

[Hl] E. Hlawka, Theorie der Gleichverteilung, Bibliographisches Institut, Mannheim, 1979. | MR | Zbl

[KN] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, J. Wiley, New York, 1974. | MR | Zbl

[Od] A.M. Odlyzko, Enumeration of Strings, in Combinatorial Algorithms on Words, A. Apostolico and Z. Galil eds., Springer, Berlin, Heidelberg New York, 1984.

[Wa] P. Walters, An Introduction to Ergodic Theory, Springer Verlag, New York, 1982. | MR | Zbl

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