Let $Y_{1}, Y_{2}, \ldots$ be i.i.d. random variables with values in $\{1,2, \ldots\}$ and $\mathbb{E}\left[Y_{1}\right]=\mu<\infty$. Moreover, suppose that the greatest common divisor of $\left\{n: \mathbb{P}\left(Y_{1}=n\right)>0\right\}$ is 1 . Consider the following process

$$X_{n}=\inf \left\{m \geqslant n: Y_{1}+\ldots+Y_{k}=m, \text { for some } k \geqslant 0\right\}-n .$$

(a) Show that $X$ is a Markov chain and find its transition probabilities.

(b) Let $T_{0}=\inf \left\{n \geqslant 1: X_{n}=0\right\}$. Find $\mathbb{E}_{0}\left[T_{0}\right]$.

(c) Find the limit as $n \rightarrow \infty$ of $\mathbb{P}\left(X_{n}=0\right)$. State carefully any theorems from the course that you are using.