I want to believe that this is true because there are no absorbing states, but it could be false. Let's not forget Pólya's theorem: random walks on integer lattices are recurrent at dimension 1 and 2 (always come back to the origin) and transient in any higher dimension (eventually never come back).

But the number of states for the wealth distribution problem isn't infinite, hm, because the total amount of money never changes. Yeah, I think I just convinced myself that you may be right, no state is absorbent so they must all be positive recurrent.

Maybe that's how we can restore our intuition on this problem. Everyone should be wealthy and poor, but it may take very long time to see the wealth flip around. While we wait for that to happen, we'll see gross wealth inequality.