That subject might not be quite accurate, but let me clarify.
At discrete times t=1,2,..., with probability 1 events of type X and Y produced by independent random processes happen infinitely often, but the expected gaps between any two Xs or Ys is nonfinite. Is it true, and if so how might one prove, that with probability 1 some X and Y will occur simultaneously?
(I'm wondering if the proof that one-dimensional simple random walks infinitely often return can transfer to two dimensions by some general principle.)
Thanks!
While it is true that a simple two-dimensional random walk returns to the start with probability $1$ and the expected number of returns is infinite, despite the infinite expected return time for a simple one-dimensional random walk, this is not the case for higher dimensional random walks. See the earlier question Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1 for discussion and proofs
So if $X$ and $Y$ are the events of a return to the start of independent two-dimensional random walks, then probability of that they ever both occur simultaneously is like the return of a four-dimensional random walk, and so less than $1$