We define $T_i$ to be the time of the first visit to the state $i$. A state $i$ is said to be recurrent if $\mathbb{P}_i(T_i<\infty)=1$ ($\mathbb{P}_i$ is the conditional probabilty corresponding to the event $\{{X_0=i}\}$). We say a state $i$ is positive recurrent if $\mathbb{E}_i(T_i)<\infty$. It is easy to see that a positive recurrent state is recurrent.
Furthermore, we define a recurrent state $i$ to be null recurrent if it $\mathbb{E}_i(T_i)=\infty$.
I have proven the following facts:
A state $i$ is recurrent if and only if $\sum\limits_{n=0}^\infty p_{ii}^{(n)}=\infty$.
I am stuck as to how to prove (or if it is event true) the following:
- Let $i$ be a state. Then $\mathbb{E}_i(T_i)=\infty$ if and only if $\lim_{n\to\infty}p_{ii}^{(n)}=0$.
- If $\mathbb{E}_i(T_i)=\infty$ then for and state $j$ we have $\lim_{n\to\infty}p_{j,i}^{(n)}=0$
I tried using the fact $T_i=\sum\limits_{n\geq 1} \chi_{T_i\geq n}$ but it did not lead anywhere.