TITLE: When do 3D random walks return to their origin?
QUESTION [25 upvotes]: The probability of a random walk returning to its origin is 1 in two dimensions (2D) but only 34% in three dimensions: This is Pólya's theorem.  I have learned that in 2D the condition of returning to the origin holds even for step-size distributions with finite variance, and
as Byron Schmuland
kindly explained in this Math.SE posting, even for distributions with infinite variance, recurrence depends upon the details of the step-length tail distribution.  But this is all in 2D.
My question is:

Are there conditions on the step-size and step-direction distributions in three dimensions  (3D) that ensure
  that the walk will return to the origin with probability 1?

Of course I exclude here a step-direction distribution that squashes 3D → 2D.
But perhaps partial dimensional compression suffices?
 (source)
(3D random-walk image credit to http://logo.twentygototen.org/.)

REPLY [2 votes]: The answer to your question appears to be contained in a 1928 Theorem of Polya, on simple random works on a lattice $\mathbb Z^d$. As a soft-reference, see page 15 of this short paper, using only very basic mathematics (Stirling's formula, etc.). The crux of the solution is to show that in $3$ dimensions, the average number of returns to the origin is finite, and so a.e you eventually get lost for good!