= Solution
{c}
The restriction $Q_C$ is a <finite Markov chain> that is an <irreducible Markov chain>. It is <aperiodic>, because its upper states have self-transition <probability> $a/(a+b)>0$, and the <period is constant on a communicating class>. We now find its <stationary distribution>.
Every column of $Q_C$ sums to one. Here is the full incoming-mass check, with each total measured in units of $1/(a+b)$. For $j\in L$, the incoming contributions are $a$ from $j+b\in I$ and $b$ from the self-transition. For $j\in U$, they are $b$ from $j-a\in I$ and $a$ from the self-transition.
For $j\in I$, the upward predecessor $j-a$ lies in $I$ exactly when $j\geq-c+a$, contributing $b$ units. When $j\leq-c+a-1$, that predecessor is replaced by resets from all $b$ states of $L$, again contributing $b$ units. Independently, the downward predecessor $j+b$ contributes $a$ units when $j\leq c-b$; when $j\geq c-b+1$, resets from the $a$ states of $U$ contribute those $a$ units instead. Thus every column total is $a+b$ units.
Therefore $Q_C$ is a <doubly stochastic matrix>, and the <uniform distribution> on $C$ is stationary:
$$
\boxed{\pi_j=\begin{cases}\displaystyle\frac1{2c+a+b+1},&-c-b\leq j\leq c+a,\\0,&\text{otherwise}.\end{cases}}
$$
The <countable-state Markov chain convergence theorem>, applied to the finite closed class $C$, gives $Q_C^n(i,j)\to\pi_j$ for every $i\in C$. Its hypotheses hold: $C$ is finite and irreducible, hence <positive recurrent>, and it is <aperiodic>.
If $i_0>c+a$, the <hitting time> $H$ of $C$ has $\mathbb P(H>n)=(a/(a+b))^n$. If $i_0<-c-b$, the corresponding tail is $(b/(a+b))^n$. Thus $H<\infty$ <almost surely> from every exterior initial state. Conditioning on $H$ and $R_H$, use convergence within $C$ after entrance and the <bounded convergence theorem>; equivalently, first restrict to $H\leq K$ and then let $K\to\infty$. This proves, for every $i_0\in\mathbb Z$,
$$
\boxed{R_n\xrightarrow{\mathrm d}\operatorname{Unif}\{-c-b,\ldots,c+a\}.}
$$
The <limiting distribution> is independent of $i_0$.
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