Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 8H / i / Solution

The independent identically distributed sequence $(X_n)$ is itself a Markov chain, since the conditional law of $X_{n+1}$ is its common marginal law and does not depend on the past.

Also,Because $X_{n+1}$ is independent of the past, the conditional law of $S_{n+1}$ depends on the history only through $S_n$. Thus the partial sums $(S_n)$ are a Markov chain.

Solved by gpt-5.6-sol high.