On the discrepancy of Markov-normal sequences

M. B. Levin

Journal de théorie des nombres de Bordeaux (1996)

  • Volume: 8, Issue: 2, page 413-428
  • ISSN: 1246-7405

Abstract

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We construct a Markov normal sequence with a discrepancy of O ( N - 1 / 2 log 2 N ) . The estimation of the discrepancy was previously known to be O ( e - c ( log N ) 1 / 2 ) .

How to cite

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Levin, M. B.. "On the discrepancy of Markov-normal sequences." Journal de théorie des nombres de Bordeaux 8.2 (1996): 413-428. <http://eudml.org/doc/247824>.

@article{Levin1996,
abstract = {We construct a Markov normal sequence with a discrepancy of $O(N^\{- 1/2\} \log ^2 N)$. The estimation of the discrepancy was previously known to be $O(e^\{- c (\log N)^\{1/2\}\})$.},
author = {Levin, M. B.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {uniform distribution mod 1; discrepancy; Markov normal sequence; irreducible Markov transition matrix},
language = {eng},
number = {2},
pages = {413-428},
publisher = {Université Bordeaux I},
title = {On the discrepancy of Markov-normal sequences},
url = {http://eudml.org/doc/247824},
volume = {8},
year = {1996},
}

TY - JOUR
AU - Levin, M. B.
TI - On the discrepancy of Markov-normal sequences
JO - Journal de théorie des nombres de Bordeaux
PY - 1996
PB - Université Bordeaux I
VL - 8
IS - 2
SP - 413
EP - 428
AB - We construct a Markov normal sequence with a discrepancy of $O(N^{- 1/2} \log ^2 N)$. The estimation of the discrepancy was previously known to be $O(e^{- c (\log N)^{1/2}})$.
LA - eng
KW - uniform distribution mod 1; discrepancy; Markov normal sequence; irreducible Markov transition matrix
UR - http://eudml.org/doc/247824
ER -

References

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  1. [1] A.G. Postnikov and I.I. Piatetski-Shapiro, A Markov sequence of symbols and a normal continued fraction, Izv. Akad. Nauk SSSR, Ser. Mat., 1957, v.21, 501-514. Zbl0078.31102MR101856
  2. [2] M. Smorodinsky and B. Weiss, Normal sequences for Markov shifts and intrinsically ergodic subshifts, Israel Journal of Mathematics, 1987, v.59, 225-233. Zbl0643.10041MR920084
  3. [3] A. Bertrand-Mathis, Points generiques de Champernowne sur certains systems codes; application aux θ-shifts, Ergod. Th. & Dynam. Sys.,1988, v. 8, 35-51. Zbl0657.28014
  4. [4] N.N. Chentsov, Pseudorandom numbers for modeling Markov chains, U.S.S.R. Comput. Maths. Math. Phis., 1967, vol. 7, no 3, 218-233. Zbl0181.45105
  5. [5] N.M. Korobov, Exponential sums and their applications, Dordrecht, 1992, 209 pages. Zbl0754.11022MR1162539
  6. [6] U.N. Sahov, Imitation of simplist Markov processes, Izv. Akad. Nauk SSSR, Ser. Mathem., 1959, v.23, 815-822. MR114241
  7. [7] U.N. Sahov, The construction of sequence of signs that is normal in the sen se of Markov, Moskovskii Gosudarstvennyi Pedagogiceskii institute im. V.I. Lenina, Ucenye Zapiski, 1971, v. 375,143-155. 
  8. [8] W. Feller, An Introduction to Probability Theory and Its Applications, vol.1, New York, 1965. Zbl0039.13201
  9. [9] J.L. Kemeny and J.L. Snell, Finite Markov chains, New York, 1960, 210 pages. Zbl0089.13704MR115196
  10. [10] I.S. Berezin and N.P. Zhidkov, Computing methods, vol. 2, Pergamon Press, Oxford, 1965, 267, 268. Zbl0122.12903
  11. [11] N.M. Korobov, Distribution of fractional parts of exponential function, Vestnic Moskov. Univ.,Ser.1 Mat. Meh., 1966, v. 21, no. 4, 42-46. Zbl0154.04801MR197435
  12. [12] M.B. Levin, The distribution of fractional parts of the exponential function, Soviet. Math. (Iz. Vuz.), 1977, v. 21, no.11, 41-47. Zbl0389.10037MR506058
  13. [13] U.N. Sahov, Some bounds in the construction of Bernoulli-normal sequences of signs, Math. Notes, 1971, v. 10, 724-730. Zbl0248.65071
  14. [14] M.B. Levin, On normal sequence for Markov and Bernoulli shifts, 49-53, Proccedings of the Israel Mathematical Union Conference,1994, Beer Sheva, 97-100. 
  15. [15] W. Philipp, Limit theorems for lacunary series and uniform distribution mod 1, Acta Arithm., 1975, v. 26, 241-251. Zbl0263.10020MR379420

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