On the limit distribution of the well-distribution measure of random binary sequences

Christoph Aistleitner[1]

  • [1] TU Graz, Department of Analysis and Computational Number Theory (Math A), Steyrergasse 30/II 8010 Graz, Austria

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 2, page 245-259
  • ISSN: 1246-7405

Abstract

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We prove the existence of a limit distribution of the normalized well-distribution measure W ( E N ) / N (as N → ∞ ) for random binary sequences E N , by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.

How to cite

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Aistleitner, Christoph. "On the limit distribution of the well-distribution measure of random binary sequences." Journal de Théorie des Nombres de Bordeaux 25.2 (2013): 245-259. <http://eudml.org/doc/275695>.

@article{Aistleitner2013,
abstract = {We prove the existence of a limit distribution of the normalized well-distribution measure $W(E_N)/\sqrt\{N\}$ (as $N \rightarrow \infty $) for random binary sequences $E_N$, by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.},
affiliation = {TU Graz, Department of Analysis and Computational Number Theory (Math A), Steyrergasse 30/II 8010 Graz, Austria},
author = {Aistleitner, Christoph},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {9},
number = {2},
pages = {245-259},
publisher = {Société Arithmétique de Bordeaux},
title = {On the limit distribution of the well-distribution measure of random binary sequences},
url = {http://eudml.org/doc/275695},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Aistleitner, Christoph
TI - On the limit distribution of the well-distribution measure of random binary sequences
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/9//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 2
SP - 245
EP - 259
AB - We prove the existence of a limit distribution of the normalized well-distribution measure $W(E_N)/\sqrt{N}$ (as $N \rightarrow \infty $) for random binary sequences $E_N$, by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.
LA - eng
UR - http://eudml.org/doc/275695
ER -

References

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  1. N. Alon, Y. Kohayakawa, C. Mauduit, C. G. Moreira, and V. Rödl, Measures of pseudorandomness for finite sequences: minimal values. Combin. Probab. Comput. 15(1-2) (2006), 1–29. Zbl1084.11043MR2195573
  2. N. Alon, Y. Kohayakawa, C. Mauduit, C. G. Moreira, and V. Rödl. Measures of pseudorandomness for finite sequences: typical values. Proc. Lond. Math. Soc. (3), 95(3) (2007), 778–812. Zbl1124.68084MR2368283
  3. N. Alon, S. Litsyn, and A. Shpunt. Typical peak sidelobe level of binary sequences. IEEE Trans. Inform. Theory, 56(1) (2010), 545–554. MR2589463
  4. I. Berkes, W. Philipp, and R. F. Tichy. Empirical processes in probabilistic number theory: the LIL for the discrepancy of ( n k ω ) mod 1 . Illinois J. Math., 50(1-4) (2006), 107–145. Zbl1145.11058MR2247826
  5. I. Berkes, W. Philipp, and R. F. Tichy. Pseudorandom numbers and entropy conditions. J. Complexity, 23(4-6) (2007), 516–527. Zbl1133.11045MR2372011
  6. P. Billingsley. /it Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition (1999). Zbl0172.21201MR1700749
  7. J. Cassaigne, C. Mauduit, and A. Sárközy. On finite pseudorandom binary sequences. VII. The measures of pseudorandomness. Acta Arith., 103(2) (2002), 97–118. Zbl1126.11330MR1904866
  8. W. Feller. The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Statistics, 22 (1951), 427–432. Zbl0043.34201MR42626
  9. P. Hubert, C. Mauduit, and A. Sárközy. On pseudorandom binary lattices. Acta Arith., 125(1) (2006), 51–62. Zbl1155.11044MR2275217
  10. Y. Kohayakawa, C. Mauduit, C. G. Moreira, and V. Rödl. Measures of pseudorandomness for finite sequences: minimum and typical values. In Proceedings of WORDS’03, volume 27 of TUCS Gen. Publ., (2003), 159–169. Turku Cent. Comput. Sci., Turku. Zbl1183.11043MR2081349
  11. C. Mauduit and A. Sárközy. On finite pseudorandom binary sequences. I. Measure of pseudorandomness, the Legendre symbol. Acta Arith., 82(4) (1997), 365–377. Zbl0886.11048MR1483689
  12. A. W. van der Vaart and J. A. Wellner. Weak convergence and empirical processes. Springer Series in Statistics. Springer-Verlag, New York, (1996). Zbl0862.60002MR1385671

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