For simplicity, let $d = 2$ and consider the points $(n, m)$ in $Z^2_+$, with $\theta m \leq n \leq \theta^{-1}m$, where $0 < \theta < 1$. For i.i.d. random variables with this set as an index set we present a law of the iterated logarithm, strong laws of large numbers and related results. We also observe that (and try to explain why) the martingale proof of the Kolmogorov strong law of large numbers yields a weaker result for this index set than the classical proofs, whereas this is not the case if the index set is all of $Z^d_+, d \geq 1$.