Natural divisors and the brownian motion

Eugenijus Manstavičius

Journal de théorie des nombres de Bordeaux (1996)

  • Volume: 8, Issue: 1, page 159-171
  • ISSN: 1246-7405

Abstract

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A model of the Brownian motion defined in terms of the natural divisors is proposed and weak convergence of the related measures in the space 𝐃 [0,1] is proved. An analogon of the Erdös arcsine law, known for the prime divisors [6] (see [14] for the proof), is obtained. These results together with the author’s investigation [15] extend the systematic study [9] of the distribution of natural divisors. Our approach is based upon the functional limit theorems of probability theory.

How to cite

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Manstavičius, Eugenijus. "Natural divisors and the brownian motion." Journal de théorie des nombres de Bordeaux 8.1 (1996): 159-171. <http://eudml.org/doc/247819>.

@article{Manstavičius1996,
abstract = {A model of the Brownian motion defined in terms of the natural divisors is proposed and weak convergence of the related measures in the space $\mathbf \{D\}$ [0,1] is proved. An analogon of the Erdös arcsine law, known for the prime divisors [6] (see [14] for the proof), is obtained. These results together with the author’s investigation [15] extend the systematic study [9] of the distribution of natural divisors. Our approach is based upon the functional limit theorems of probability theory.},
author = {Manstavičius, Eugenijus},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {arithmetical functions; Brownian motion; random variables; random process; Wiener process; arcsine laws; divisors},
language = {eng},
number = {1},
pages = {159-171},
publisher = {Université Bordeaux I},
title = {Natural divisors and the brownian motion},
url = {http://eudml.org/doc/247819},
volume = {8},
year = {1996},
}

TY - JOUR
AU - Manstavičius, Eugenijus
TI - Natural divisors and the brownian motion
JO - Journal de théorie des nombres de Bordeaux
PY - 1996
PB - Université Bordeaux I
VL - 8
IS - 1
SP - 159
EP - 171
AB - A model of the Brownian motion defined in terms of the natural divisors is proposed and weak convergence of the related measures in the space $\mathbf {D}$ [0,1] is proved. An analogon of the Erdös arcsine law, known for the prime divisors [6] (see [14] for the proof), is obtained. These results together with the author’s investigation [15] extend the systematic study [9] of the distribution of natural divisors. Our approach is based upon the functional limit theorems of probability theory.
LA - eng
KW - arithmetical functions; Brownian motion; random variables; random process; Wiener process; arcsine laws; divisors
UR - http://eudml.org/doc/247819
ER -

References

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  1. [1] G.J. Babu, Probabilistic methods in the theory of arithmetic functions, Ph.D. dissertation, The Indian Statistical Institute, Calcutta, (1973). 
  2. [2] P. Billingsley, Convergence of Probability Measures, Wiley & Sons, New York, (1968). Zbl0172.21201MR233396
  3. [3] P. Billingsley, Additive functions and Brownian motion, Notices Amer. Math. Soc.17 (1970), 1050. 
  4. [4] P. Billingsley, The probability theory of additive arithmetic functions, The Annals of Probab. 2 No 5 (1974), 749-791. Zbl0327.10055MR466055
  5. [5] J.-M. Deshouillers, F. Dress & G. Tenenbaum, Lois de répartition des diviseurs, 1, Acta Arithm34 (1979), 273-285. Zbl0408.10035MR543201
  6. [6] P. Erdös, On the distribution of prime divisors, Aequationes Math.2 (1969), 177-183. Zbl0174.08104MR244178
  7. [7] P. Erdös, M. Kac, On the number of positive sums of independent random variables, Bull. of the American Math. Soc.53 (1947), 1011-1020. Zbl0032.03502MR23011
  8. [8] B. Grigelionis, R. Mikulevičius, On the functional limit theorems of the probabilistic number theory, Lithuanian Math.J.24 No 2 (1984), 72-81, (Russian). Zbl0565.10051MR773596
  9. [9] R.R. Hall, G. Tenenbaum, Divisors, Cambridge University PressCambridge (1988). Zbl0653.10001MR964687
  10. [10] J. Kubilius, Probabilistic Methods in the Theory of NumbersTransl. Math. Monographs, Amer.Math.Soc., Providence, R.I.11 (1964). Zbl0133.30203MR160745
  11. [11] M. Loève, Probabality Theory, D. van Nostrand Company, New York (1963), (3rd edition). Zbl0108.14202MR203748
  12. [12] E. Manstavičius, Arithmetic simulation of stochastic processes, Lithuanian Math. J.24 No 3 (1984)), 276-285. Zbl0565.10052
  13. [13] E. Manstavičius, An invariance principle for additive arithmetic functions, Soviet Math. Dokl.37 No 1 (1988), 259-263. Zbl0689.10057MR947794
  14. [14] E. Manstavičius, "Probability Theory and Mathematical Statistics. Proceedings of the Sixth Vilnius Conference" (1993) B.Grigelionis et al Eds) VSP/TEV, (1994), 533-539. Zbl0849.11067MR1649597
  15. [15] E. Manstavičius, Functional approach in the divisor distribution problems, Acta Math. Hungarica66 No 3 (1995), 343-359. Zbl0823.11045MR1316707
  16. [16] W. Philipp, Arithmetic functions and Brownian motion, Proc. Sympos. Pure Math.24 (1973), 233-246. Zbl0269.10031MR354602
  17. [17] I.Z. Ruzsa, Effective results in probabilistic number theory, In, Théorie élémentaire et analytique des nombres ed. J.Coquet, 107-130, Dépt. Math. Univ.Valenciennes,. 
  18. [18] G. Tenenbaum, Lois de répartition des diviseurs, 4, Ann. Inst. Fourier29 (1979), 1-15. Zbl0403.10029MR552957
  19. [19] N.M. Timofeev, Ch.Ch. Usmanov, On arithmetical modelling of the Brownian motion, Dokl. Acad. Sci. Tadz.SSR25 No 4 (1982), 207-211, (Russian). Zbl0504.10030MR686370
  20. [20] N.M. Timofeev, Ch.Ch. Usmanov, On arithmetical modelling of random processes with independent increments, Dokl. Acad. Sci. TadzSSR27 No 10 (1984), 556-559, (Russian). Zbl0577.10044MR784689

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