I have an irreducible transient Markov chain with infinite state space {0,1,2,...} and I am trying to find the probability that the chain never returns to the initial state (let's say state 0). For example, consider an asymmetric random walk, in which the states (starting from zero) have a different probability each, so the probability of going left $p_{i,i-1}$ is less than the probability of going right $p_{i,i+1}$, but when the step n goes to infinity, both probabilities are the same (1/2).
How can I prove that the chain never returns to zero?
A state is transient if there's a nonzero probability that, starting there, that state is never visited again.
In other words: $f_{ii}=P(X_n=i\ for\ some\ n \ge 1|X_0=i)>0$ In this case $i=0$
Thank you.
This question is false as stated. If $p_{0,1}=1$, $p_{i,i+1}=\frac{i+1}{2i+1}$ and $p_{i,i-1}=\frac{i}{2i+1}$, then here I show that the chain is in fact recurrent.