  We study the path behaviour of a simple random walk on the 2-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.
