Block distribution in random strings

Grabner, Peter J.

doi:10.5802/aif.1345

Grabner, Peter J.

Annales de l'Institut Fourier, Tome 43 (1993) no. 2, pp. 539-549.

Résumé
Abstract

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 $\log_{\frac{1}{p}} N - \log_{\frac{1}{p}} N - ψ (N)$ (avec $p \leq q$ ) où $ψ (N)$ tend vers $+ \infty$ . Ce résultat généralise celui de P. Flajolet, P. Kirschenhofer et R.F. Tichy concernant le cas uniforme $p = q = \frac{1}{2}$ .

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 $\log_{\frac{1}{p}} N - \log_{\frac{1}{p}} N - ψ (N)$ (for $p \leq q$ ) where $ψ (N)$ tends to $+ \infty$ . This generalizes a result due to P. Flajolet, P. Kirschenhofer and R.F. Tichy concerning the case $p = q = \frac{1}{2}$ .

MR Zbl

DOI : 10.5802/aif.1345

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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},
     mrnumber = {94d:60044},
     zbl = {0778.60023},
     language = {en},
     url = {https://www.numdam.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://www.numdam.org/articles/10.5802/aif.1345/

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

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