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

@article{AIF_1993__43_2_539_0,
     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 = {http://www.numdam.org/articles/10.5802/aif.1345/}
}

TY  - JOUR
AU  - Grabner, Peter J.
TI  - Block distribution in random strings
JO  - Annales de l'Institut Fourier
PY  - 1993
SP  - 539
EP  - 549
VL  - 43
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1345/
DO  - 10.5802/aif.1345
LA  - en
ID  - AIF_1993__43_2_539_0
ER  -

%0 Journal Article
%A Grabner, Peter J.
%T Block distribution in random strings
%J Annales de l'Institut Fourier
%D 1993
%P 539-549
%V 43
%N 2
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1345/
%R 10.5802/aif.1345
%G en
%F AIF_1993__43_2_539_0

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. http://www.numdam.org/articles/10.5802/aif.1345/

Bibliographie
Cité par

[Fe] W. Feller, An Introduction to Probability Theory and Its Applications, J. Wiley, New York, 1965.

[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

Cité par Sources :