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

   
@article{EJP16,
	author = {Jay Rosen},
	title = {Laws of the Iterated Logarithm for Triple Intersections of Three Dimensional Random Walks},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {2},
	year = {1997},
	keywords = {random walks, intersections},
	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 $n$.},
	pages = {no. 2, 1-32},
	issn = {1083-6489},
	doi = {10.1214/EJP.v2-16},    
        url = {http://ejp.ejpecp.org/article/view/16}}