Htykuut

I am trying to get some intuition on Harris recurrence in Markov chains. Define state space $mathcal S$ comprising a single communication class, $f_{ii}^{(n)}=P(X_n=i, X_{n-1}ne i,ldots X_1ne imid X_0=i)$, $f_{ii}=sum_n f_{ii}^{(n)}$, $T_{ii}=inf_n {X_n=imid X_0=i}$ and $E(T_{ii})=sum_n nf_{ii}^{(n)}$ and $V_i=sum_nmathbb 1_{X_n=i}$, we have the following.

• 
  Transience: $f_{ii}<1$


• 
  Null recurrence:$f_{ii}=1$, $E(T_{ii})=infty$


• 
  Positive recurrence: $f_{ii}=1$, $E(T_{ii})<infty$, $E(V_i)=infty forall iin mathcal S$


• 
  Harris recurrence: $f_{ii}=1$, $P(omega:V_i(omega)=infty)=1 forall iin mathcal S$

Are the above relations correct? I do not see how the last bullet relates to the definition in Wikipedia. Are there any examples of finite Markov chains that are positive but not Harris recurrent?

This answer may be wrong, but I think it is worth posting and if it is wrong, someone can point it out and I can learn something too.

I think you do not mean a finite Markov Chain, because for a finite state chain, assuming it is irreducible, every state will be visited infinitely often, there is no question.

I think there is only a difference if the state space is uncountable.

This is because the event $V_i=infty$ is the same as the event "state i is visited infinitely often". This has probability 1 or 0, by Levy's zero-one law.

So, suppose a positive definite chain is not Harris recurrent. This means the expected number of visits to $i$ is infinite, but the number visits to $i$ is finite, almost surely, but doesn't this mean it is transient?