Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
On the character of convergence to Brownian local time. II
Download PDF
Download PDF
  • Published: June 1986

On the character of convergence to Brownian local time. II

  • A. N. Borodin1 

Probability Theory and Related Fields volume 72, pages 251–277 (1986)Cite this article

  • 229 Accesses

  • 23 Citations

  • Metrics details

Summary

In this paper we consider the sequences of stochastic processes which converge weakly asn→∞ to Brownian local time. These processes are generated by a recurrent random walk with finite variance. The main result is the following: it is possible to redefine a random walk in such a way that for a wide class of processes the normalized differences between them and Brownian local time converge in distribution to some stochastic process. We also prove that such differences with probability one have the logarithmic upper bound. It is so called “Strong invariance principles for local times”.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Aleskeviciene, A.K.: On asymptotic distribution of local times of a recurrent random walk. In: Abstracts of Communications of IV USSR-Japan Symposium on Probability Theory and Mathematical Statistics vol.1, pp. 97–98. Tbilisi: Metsniereba 1982

    Google Scholar 

  2. Borodin, A.N.: An asymptotic behaviour of local times of a recurrent random walk with finite variance. Theory Probab. Appl.26, 769–783 (1981)

    Google Scholar 

  3. Borodin, A.N.: On distribution of integral type functionals of Brownian motion. Zapiski Nauchnych Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V.A.Steklova AN SSSR119, 19–38 (1982)

    Google Scholar 

  4. Borodin, A.N.: On the character of convergence to Brownian local time. Dokl. USSR Academy of Sciences269, 784–788 (1983)

    Google Scholar 

  5. Borodin, A.N.: On distribution of random walk local time. LOMI Preprint E-4-84. Leningrad 1984

  6. Borodin, A.N.: On the character of convergence to Brownian local time I. Prob. Th. Rel. Fields72, 231–250 (1986)

    Google Scholar 

  7. Csáki, E., Révész, P.: Strong Invariance for local times. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 263–278 (1983)

    Google Scholar 

  8. Dobrushin, R.L.: Two limit theorems for simplest random walk on a line. Usp. Mat. Nauk10, 139–146 (1955)

    Google Scholar 

  9. Dobrushin, R.L.: The continuity condition for sample martingale functions. Theory Probab. Appl.3, 97–98 (1958)

    Google Scholar 

  10. Doob, J.L.: Stochastic processes, New York: Wiley 1953

    Google Scholar 

  11. Ito, K., McKean, H.P.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965

    Google Scholar 

  12. Kesten, H.: An iterated logarithm law for the local time. Duke Math. J.32, 447–456 (1965)

    Google Scholar 

  13. Knight, F.B.: Random walks and a sojourn density process of Brownian motion. Trans. Am. Math. Soc.109, 56–86 (1963)

    Google Scholar 

  14. McKean, H.P.: Stochastic integrals. New York-London: Academic Press 1969

    Google Scholar 

  15. Perkins, E.: Weak invariance principles for local time. Z. Wahrscheinlichkeitstheor. Verw. Geb.60,437–451 (1982)

    Google Scholar 

  16. Ray, D.B.: Sojourn times of a diffusion process. Ill. J. Math.7, 615–630 (1963)

    Google Scholar 

  17. Révész, P.: Local time and invariance. Lecture Notes in Math.861. Berlin-Heidelberg-New York: Springer 1981

    Google Scholar 

  18. Révész, P: A strong invariance principle of the local time of R.V.'s with continuous distribution. Stud. Sci. Math. Hung.16, 219–228 (1981)

    Google Scholar 

  19. Skorokhod A.V., Slobodenyuk, N.P.: Limit theorems for random walks. Kiev: Naukova Dumka 1970

    Google Scholar 

  20. Trotter, H.F.: A property of Brownian motion paths. Ill. J. Math.2, 425–433 (1958)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Leningrad Branch Steklov Institute of Mathematics Academy of Sciences of the USSR, Fontanka 27, 191011, Leningrad, USSR

    A. N. Borodin

Authors
  1. A. N. Borodin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Borodin, A.N. On the character of convergence to Brownian local time. II. Probab. Th. Rel. Fields 72, 251–277 (1986). https://doi.org/10.1007/BF00699106

Download citation

  • Received: 25 June 1984

  • Accepted: 11 November 1985

  • Issue Date: June 1986

  • DOI: https://doi.org/10.1007/BF00699106

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Process
  • Random Walk
  • Probability Theory
  • Local Time
  • Mathematical Biology
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature