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 Electronic Journal of Probability > Vol. 2 (1997) > Paper 2 open journal systems 


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.


Full text: PDF

Pages: 1-32

Published on: March 26, 1997
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Electronic Journal of Probability. ISSN: 1083-6489