Laws of the iterated logarithm for triple intersections of three dimensional random walks. (English) Zbl 0890.60065

Rosen, Jay

Electron. J. Probab. 2, Paper 2, 32 p. (1997).

Summary: 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. 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 \(n\).

MSC:

60G50	Sums of independent random variables; random walks
60F99	Limit theorems in probability theory

Keywords:

random walks; intersections