Consider a Markov chain with state space $S=\{0,1,2, \ldots\}$ and transition matrix given by

$$P_{i, j}= \begin{cases}q p^{j-i+1} & \text { for } i \geqslant 1 \text { and } j \geqslant i-1 \\ q p^{j} & \text { for } i=0 \text { and } j \geqslant 0\end{cases}$$

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

For each value of $p, 0<p<1$, determine whether the chain is transient, null recurrent or positive recurrent, and in the last case find the invariant distribution.