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

@inproceedings{Rosen1998LawsOT,
title={Laws of the Iterated Logarithm for Triple Intersections of Three Dimensional Random Walks},
author={Jay Rosen},
year={1998}
}

Let X = {Xn, n ≥ 1}, X ′ = {X ′ n, n ≥ 1} and X ′′ = {X ′′ n, n ≥ 1} be three independent copies of a symmetric random walk in Z3 with E(|X1| log+ |X1|) < ∞. In this paper we study the asymptotics of In, the number of triple intersections up to step n of the paths of X , X ′ and X ′′ as n→∞. Our main result is lim sup n→∞ In log(n) log3(n) = 1 π|Q| a.s. where Q denotes the covariance matrix of X1. A similar result holds for Jn, the number of points in the triple intersection of the ranges of X… CONTINUE READING