Consider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ with state space $\{0,1,2, \ldots\}$ and transition probabilities given by

$$P_{i, j}=p q^{i-j+1}, \quad 0<j \leqslant i+1, \quad \text { and } \quad P_{i, 0}=q^{i+1} \quad \text { for } \quad i \geqslant 0$$

with $P_{i, j}=0$, otherwise, where $0<p<1$ and $q=1-p$.

For each $i \geqslant 1$, let

$$h_{i}=\mathbb{P}\left(X_{n}=0, \text { for some } n \geqslant 0 \mid X_{0}=i\right),$$

that is, the probability that the chain ever hits the state 0 given that it starts in state $i$. Write down the equations satisfied by the probabilities $\left\{h_{i}, i \geqslant 1\right\}$ and hence, or otherwise, show that they satisfy a second-order recurrence relation with constant coefficients. Calculate $h_{i}$ for each $i \geqslant 1$.

Determine for each value of $p, 0<p<1$, whether the chain is transient, null recurrent or positive recurrent and in the last case calculate the stationary distribution.

[Hint: When the chain is positive recurrent, the stationary distribution is geometric.]