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Strong limit theorems for a simple random walk on the 2-dimensional comb. (English) Zbl 1190.60020

Summary: We study the path behaviour of a simple random walk on the two-dimensional comb lattice \(\mathbb C^2\) that is obtained from \(\mathbb Z^2\) by removing all horizontal edges off the \(x\)-axis. In particular, we prove a strong approximation result for such a random walk which, in turn, enables us to establish strong limit theorems, like the joint Strassen type law of the iterated logarithm of its two components, as well as their marginal Hirsch type behaviour.

MSC:

60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
60G50 Sums of independent random variables; random walks
60J65 Brownian motion
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