Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T23:15:34.857Z Has data issue: false hasContentIssue false
  • English
  • Français

Arithmetical representations of Brownian motion I

Published online by Cambridge University Press:  12 March 2014

Willem Fouché
Willem Fouché*
Affiliation:
Department of Mathematics and Applied Mathematics, University of Pretoria, 0002 Pretoria, South Africa, E-mail: wlfouche@math.up.ac.za
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss ways in which a typical one-dimensional Brownian motion can be approximated by oscillations which are encoded by finite binary strings of high descriptive complexity. We study the recursive properties of Brownian motions that can be thus obtained.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 1 , March 2000 , pp. 421 - 442
Copyright
Copyright © Association for Symbolic Logic 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Asarin, E. A. and Prokovskiy, A. V., Primeenenie kolmogorovskoi slozhnosti k anlizu dinamiki upravlemykh sistem, Automatika i Telemekhanika, vol. 1 (1986), pp. 25–33.Google Scholar
[2]Billingsley, P., Convergence of probability measures, John Wiley and Sons, New York, 1968.Google Scholar
[3]Billingsley, P., Probability and measure, second ed., John Wiley and Sons, New York, 1995.Google Scholar
[4]Chaitin, G. A., On the length of programs for computing binary sequences, Journal of the Association for Compututing Machinery, vol. 13 (1966), pp. 547–569.CrossRefGoogle Scholar
[5]Chaitin, G. A., Algorithmic information theory, Cambridge University Press, 1987.CrossRefGoogle Scholar
[6]Donsker, M. D., An invariance principle for certain probability limit theorems, Memoirs of the American Mathematical Society, vol. 6 (1951).Google Scholar
[7]Fouché, W. L., Descriptive complexity and reflective properties of combinatorial configurations, Journal of the London Mathematical Society, vol. 54 (1996), pp. 199–208.CrossRefGoogle Scholar
[8]Freedman, D., Brownian motion and diffusion, Holden-Day, 1971.Google Scholar
[9]Hida, T., Brownian motion, Springer-Verlag, New York, 1980.CrossRefGoogle Scholar
[10]Hinman, P. G., Recursion-theoretic hierarchies, Springer-Verlag, New York, 1978.CrossRefGoogle Scholar
[11]Kahane, J.-P., Some random series of functions, second ed., Cambridge University Press, 1993.Google Scholar
[12]Khintchine, A. I., Über dyadische Brüche, Mathematische Zeitschrift, vol. 18 (1923), pp. 109–116.CrossRefGoogle Scholar
[13]Kolmogorov, A. N., Über das Gesetz der iterierten Logaritmus, Mathematische Annalen, vol. 101 (1929), pp. 126–135.CrossRefGoogle Scholar
[14]Kolmogorov, A. N., Three approaches to the quantitative definition of randomness, Problems of Information Transmission, vol. 1 (1965), pp. 1–7.Google Scholar
[15]Martin-Löf, P., The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602–619.CrossRefGoogle Scholar
[16]Shen, A. Kh., Connections between different algorithmic definitions of randomness, Soviet Math. Dokl., vol. 38 (1989), pp. 316–319.Google Scholar
[17]Simon, B., Functional integration and quantum physics, Academic Press, 1979.Google Scholar
[18]Solovay, R., Draft of a paper on Chaitin's work … done for the most part during the period of Sept.–Dec. 1974, unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, 05 1975, 215 pp.Google Scholar
[19]Uspensky, V. A. and Semenov, A. L., What are the gains of a theory of algorithms, Algorithms in modern mathematics and computer science, Lecture Notes in Computer Science, no. 122, Springer-Verlag, New York, 1981.Google Scholar
[20]van Lambalgen, M., Von Mises' definition of random sequences reconsidered, this Journal, vol. 52 (1987), pp. 725–755.Google Scholar
[21]Vitányi, P. and Li, M., An introduction to Kolmogorov complexity and its applications, Springer-Verlag, New York, 1993.Google Scholar
[22]Vov'k, V. G., The law of the iterated logarithm for Kolmogorov random or chaotic sequences, Theory of Probability audits Applications, vol. 32 (1987), pp. 413–425.CrossRefGoogle Scholar