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1997 Laws of the Iterated Logarithm for Triple Intersections of Three Dimensional Random Walks
Jay Rosen
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Electron. J. Probab. 2: 1-32 (1997). DOI: 10.1214/EJP.v2-16

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 $n$.

Citation

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Jay Rosen. "Laws of the Iterated Logarithm for Triple Intersections of Three Dimensional Random Walks." Electron. J. Probab. 2 1 - 32, 1997. https://doi.org/10.1214/EJP.v2-16

Information

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

zbMATH: 0890.60065
MathSciNet: MR1444245
Digital Object Identifier: 10.1214/EJP.v2-16

Subjects:
Primary: 60J15
Secondary: 60F99

JOURNAL ARTICLE
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Vol.2 • 1997
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