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Laws of the iterated logarithm for intersections of random walks on \(Z^ 4\). (English) Zbl 0870.60065
Summary: Let \(X=\{X_n, n\geq 1\}\), \(X'= \{X_n', n\geq 1\}\) be two independent copies of a symmetric random walk in \(Z^4\) with finite third moment. We study the asymptotics of \(I_n\), the number of intersections up to step \(n\) of the paths of \(X\) and \(X'\) as \(n\to\infty\). Our main result is \[ \limsup{I_n\over\log(n)\log_2(n)}= {1\over 2\pi^2|Q|^{1/2}}\quad\text{a.s.}, \] where \(Q\) denotes the covariance matrix of \(X_1\). A similar result holds for \(J_n\), the number of points in the intersection of the ranges of \(X\) and \(X'\) up to step \(n\).

60G50 Sums of independent random variables; random walks
60F15 Strong limit theorems
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