How to prove that each states of this Markov Chain is transient?

Question

Suppose there is a markov chain with the transition probabilities: $p_{0,2}=1,p_{i,i+2}=p_{i,i-1}=1/2,i=1,2,\ldots$. For this problem, how to prove that every state is transient?

Answer 1

I think you can do it by induction. For $i=0$, the state is transient. For $i\geq1$, bare in mind that$$\sum_{n=0}^{\infty}p_{i,i+2}^{(n)}\quad \text{and}\quad \sum_{n=0}^{\infty}p_{i,i-1}^{(n)}$$, which are absolutely convergent sums, finitely tends to zero. For any state $j$, $\displaystyle\sum_{n=0}^{\infty}p_{j,j}^{(n)}$ is just the combination which can be proven by induction. Then conclude that every state $i\geq0$ is transient.

Answer 2

Notice that $p$ is irreducible and thus if it were recurrent, then every harmonic function would be a constant. However, it's clear that the function $f:\mathbb{N} \rightarrow \mathbb{R}$ defined with $f(0)=1,f(1)=2$ and recursively defined by $pf=f$ is non-constant but harmonic. Hence, $p$ is transient.