No null recurrent state in finite state space from definition.

Question

Let $\{X_n\}$ be a markov chain on finite state space $I$, with stationary transition probabilities. Let us denote $f^n(i,i):=P(X_n=i,X_{n-1}\neq i,\ldots X_1\neq i\mid X_0=i)$. We say $i$ is recurrent if $\sum_n f^n(i,i)=1$; positive recurrent if in addition $\sum_n nf^n(i,i)<\infty$ and null recurrent if the latter sum is infinite.

Can we prove that in finite state space, there is no null recurrent state just from this definition?

I kind of get the idea that since the state space is finite, it is expected to return in finite time. But could not prove it without using other theorems (like $P^n(i,j)\to 0$).

Answer 1

Probably a bit late but yes, this can be shown quiet nicely.

1. Show that a finite Markov Chain can not be transient.
2. From (1.) we get that on the Markov chain there is at least 1 recurrent class (which can also be an absorbing state).
3. Lets observe the existing recurrent class "C". We can prove that on a recurrent class there is an existing "invariant measure" $\pi$ (which solves $\pi = \pi P $ where $ P = (p_{ij})_{i,j \in C} $ ). We can also show that $0<\pi_k<\infty$ and that the sum of the components, $ \sum\limits_{j \in C} \pi_j = \mathbb{E}_i \lbrack T \rbrack $ is equal to mean hitting time.
4. Last step. Remember that we are on a finite recurrent class. Therefore, $ \sum\limits_{j \in C} \pi_j = \mathbb{E}_i \lbrack T \rbrack < \infty $, the sum of the the finite components of our invariant measure must be finite. As the sum is the mean hitting time, the mean hitting time is finite. Therefore our recurrent class C is positive-recurrent.

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