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The moving sums $(K_n)$ are not necessarily Markov. For a counterexample, let the $X_n$ be independent Bernoulli variables with parameter $1/2$. On the eventwe must have $X_{n-1}=0$ and $X_n=1$, soOn the other hand, $K_n=1$ and $K_{n-1}=2$ force $X_{n-1}=1$ and $X_n=0$, whenceBoth conditioning events have positive probability. Knowledge of $K_n$ alone therefore does not determine the next-step law. The overlapping moving sum need not be Markov.