A Markov chain with state-space $I=\mathbb{Z}^{+}$has non-zero transition probabilities $p_{00}=q_{0}$ and

$$p_{i, i+1}=p_{i}, \quad p_{i+1, i}=q_{i+1} \quad(i \in I) .$$

Prove that this chain is recurrent if and only if

$$\sum_{n \geqslant 1} \prod_{r=1}^{n} \frac{q_{r}}{p_{r}}=\infty$$

Prove that this chain is positive-recurrent if and only if

$$\sum_{n \geqslant 1} \prod_{r=1}^{n} \frac{p_{r-1}}{q_{r}}<\infty$$