Laws of the Iterated Logarithm for Triple Intersections of Three Dimensional Random Walks
Jay Rosen, College of Staten Island, CUNY
Abstract
Let X = X_n, X' = X'_n, and X'' = X''_n, ngeq 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.
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