If state $j$ is recurrent and the following holds can it be called as positive recurrent ?
$$\lim_{n > \infty}\frac{1}{n}\sum_{k=1}^{n}p_{jj}^{(k)} > 0$$
I know that this a necessary condition for positive recurrence, but is it sufficient ?
If state $j$ is recurrent and the following holds can it be called as positive recurrent ? $$\lim_{n > \infty}\frac{1}{n}\sum_{k=1}^{n}p_{jj}^{(k)} > 0$$ I know that this a necessary condition for positive recurrence, but is it sufficient ? 


The answer is yes. A good version of the ergodic theorem for Markov Chains says that for any discrete Markov chain, $\frac1{n}\sum_{k=0}^{n1} 1_{\{X_k=j\}}$ almost surely converges to $\frac{1_{T_j<+\infty}}{E^j[T_j]}$. So, by dominated convergence, the quantity that you consider converges to $\frac{P^j(T_j<+\infty)}{E^j[T_j]}$. If the limit is positive, $E^j[T_j]<+\infty$, which is precisely positive recurrence. 

