In that post, my intuition was simply that the probability of a random walk in d dimensions being at the origin at time t scales like t<sup>-d/2</sup>, and &sum;<sub>t &gt; 0</sub> t<sup>-d/2</sup> converges if and only if d &gt; 2.  Obviously this can only be turned into a proof if <i>d</i> is an integer.