Electronic Journal of Probability

Laws of the Iterated Logarithm for Triple Intersections of Three Dimensional Random Walks

Jay Rosen

Full-text: Open access

Abstract

Let $X = X_n, X' = X'_n$, and $X'' = X''_n$, $n\geq 1$, be three independent copies of a symmetric three dimensional random walk with $E(|X_1|^{2}\log_+ |X_1|)$ finite. In this paper we study the asymptotics of $I_n$, the number of triple intersections up to step $n$ of the paths of $X, X'$ and $X''$ as $n$ goes to infinity. Our main result says that the limsup of $I_n$ divided by $\log (n) \log_3 (n)$ is equal to $1 \over \pi |Q|$, a.s., where $Q$ denotes the covariance matrix of $X_1$. A similar result holds for $J_n$, the number of points in the triple intersection of the ranges of $X, X'$ and $X''$ up to step $n$.

Article information

Source
Electron. J. Probab., Volume 2 (1997), paper no. 2, 32 pp.

Dates
Accepted: 26 March 1997
First available in Project Euclid: 26 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1453839978

Digital Object Identifier
doi:10.1214/EJP.v2-16

Mathematical Reviews number (MathSciNet)
MR1444245

Zentralblatt MATH identifier
0890.60065

Subjects
Primary: 60J15
Secondary: 60F99: None of the above, but in this section

Keywords
random walks intersections

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Rosen, Jay. Laws of the Iterated Logarithm for Triple Intersections of Three Dimensional Random Walks. Electron. J. Probab. 2 (1997), paper no. 2, 32 pp. doi:10.1214/EJP.v2-16. https://projecteuclid.org/euclid.ejp/1453839978


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References

  • Breiman, L. (1992). Probability. Corrected reprint of the 1968 original. Classics in Applied Mathematics, 7. Society for Industrial and Applied Mathematics.
  • Kahane, J.-P. (1985). Some random series of functions. Cambridge University Press.
  • Lawler, G. (1994). A note on the Green's function for random walks in four dimensions. Duke Math. Preprint, 94-03.
  • Le Gall, J.-F. (1986). Proprietes d'intersection des marches aleatoires, II. Comm. Math. Phys. 104, 509–528.
  • Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks Ann. Probab. 19, 650–705.
  • Marcus, M. and Rosen, J. Laws of the iterated logarithm for intersections of random walks on $Z^4$. Ann. Inst. H. Poincaré, Prob. Stat., to appear.
  • Marcus, M. and Rosen, J. (1994) Laws of the iterated logarithm for the local times of recurrent random walks on $Z^2$ and of Levy processes and recurrent random walks in the domain of attraction of Cauchy random variables. Ann. Inst. H. Poincaré, Prob. Stat. 30, 467–499.
  • Marcus, M. and Rosen, J. (1994) Laws of the iterated logarithm for the local times of symmetric Levy processes and recurrent random walks. Ann. Probab. 22, 626–658.
  • Spitzer, F. (1976). Principles of Random Walk. Springer Verlag