proof of convergence in arbitrary precision PRNGs
You ask us for a proof this program will always halt for all initial seeds, but no such proof is possible in general.  For instance, if the PRNG always outputs the number 66, then this will never halt (assuming you use big integers).  Obviously that would be a really lousy PRNG, but if you aren't willing to assume that the PRNG is good, then no proof is possible.

On the other hand, if you assume that the PRNG is good (i.e., computationally indistinguishable from truly random values), then this is not really a question about PRNGs.  Indeed, it's not even a question about computer science.  It is a question about whether a particular random walk will eventually return to its starting point.  (As such, this question really belongs on a mathematics site, not CS.SE.)

You ask for a proof that the random walk will never return to its starting point (that the probability of return to the initial point is exactly zero).  I do not think any such proof is possible, because I do not think the statement is actually true.