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An intersection property of the simple random walks in \(Z^ d\). (English) Zbl 0869.60065

Summary: Let \(\{S_d(n)\}_{n\geq 0}\) be the simple random walk in \(Z^d\), and \(\Pi^{(d)} (a,b)= \{S_d(n) \in Z^d: a\leq n\leq b\}\). Suppose \(f(n)\) is an integer-valued function and increases to infinity as \(n\) tends to infinity, and \(E^{(d)}_n= \{\Pi^{(d)} (0,n) \cap \Pi^{(d)} (n+f(n), \infty) \neq \emptyset\}\). A necessary and sufficient condition to ensure \(P(E_n^{(d)}\), i.o.)=0, or 1 is derived for \(d=3,4\). This problem was first studied by P. Erdös and S. J. Taylor [Acta Math. Sci. Hung. 11, 231-248 (1960; Zbl 0096.33302)].

MSC:

60G50 Sums of independent random variables; random walks

Citations:

Zbl 0096.33302
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References:

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