## Electronic Journal of Probability

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

Jay Rosen

#### 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

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

#### 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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