I'm a bit confused by this theorem, not sure of the name in english but here are the details:
Let the markov chain $X = (X_n)_{n\geq 0}$ with transition matrix $\mathbb{P}$ and state space $S$.
Let $X$ be irreducible then, if a state in $S$ is transient, then all states are transient. And every state will only be visited a finite number of times.
I'm not quite sure how a markov chain could satisfy this property. I can see how a subset of states might be transient, but to have a markov chain you need at least one recurrent state, perhaps I'm misunderstanding something.
If such a markov chain exists, then I'd like to see an example of one, I've tried searching but I can't seem to find anything on this.
Any help would be greatly appreciated.