= Solution
Write $S_{n+1}=S_n+X_{n+1}$. The new <random variable> $X_{n+1}$ is <independent> of the entire history generated by $X_1,\ldots,X_n$, hence of $(S_1,\ldots,S_n)$. This is a <Markov chain obtained from independent innovations>: conditional on the past, only the current sum enters the next state. Its time-homogeneous <transition probabilities> are
$$
\boxed{P_{st}=\begin{cases}\dfrac1{(t-s)(t-s+1)},&t>s,\\0,&t\leq s,\end{cases}\qquad s,t\in\{1,2,\ldots\}.}
$$
Every jump is a positive integer, and each row sums to one by the <probability mass function> from part (b).
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