Suppose it is said that the chain is recurrent since p(x,x-1)=1 and p(0,x)=Px

In other words, whenever the chain is at a state other than 0 it proceeds deterministically, one step at a time toward 0, and whenever the chain reaches state 0, it chooses a new state.

Under what conditions on the Px is the chain positive recurrent?

I've found the limiting distribution of x is 1/XPx.

Is it correct?

Thanks in advance!!