Let $S_n = X_1 + \dots + X_n$, where the increments $X_i$ are not i.i.d. Let $\tau$ be some random time (not necessarily a stopping time) and assume $$(\tau, X_1, \dots, X_\tau)\quad \text{ and } \quad (X_{\tau + 1}, X_{\tau + 2}, \dots) \qquad (1)$$ to be independent. I want to show with rigor that the process $(S_{\tau + n}- S_\tau)_{n \geq 1}$ is a Markov chain (satisfies the Markov property). For convenience, let us write $\tilde{S}_n = S_{\tau + n} - S_\tau$. As $\tilde{S}_n$ is independent of $\tau$ and $S_\tau$ it makes absolutely sense that the process is Markov but I would like to see the argument in more detail.
Edit: Possibly, the assumption (1) is not enough. Let us in addition to (1) assume that $$ (X_1, X_2, \dots) \overset{d}{=} (X_{\tau + 1}, X_{\tau +2}, \dots).$$ Maybe this is now enough to prove the claim that $\tilde{S}_n$ is a Markov chain.
Edit2: I am starting to think that the assumptions are not necessary. Simply by the structure of $S$ and in view of $\tilde{S}_n = X_{\tau + 1 } + \dots + X_{\tau + n}$ we have should have that $\tilde{S}_n$ conditioned on the values of $\tilde{S}_1, \dots , \tilde{S}_{n-1}$ only depends on $\tilde{S}_{n-1}$. Is this true / why is it not that easy?
I will try to answer the question myself: (I would be grateful for feedback)
By assumption (1), $S_{\tau}$ and $\tau$ are independent from $S_{\tau+ k } - S_\tau$. Thus,
\begin{align} & P(S_{\tau + n } - S_\tau \in dx \vert S_{\tau + 1} - S_\tau \in dx_1, \dots , S_{\tau + n-1} - S_\tau \in dx_{n-1}) \\ & \overset{(1)}{=} P(S_{\tau + n } - S_\tau \in dx \vert S_{\tau + 1} - S_\tau \in dx_1, \dots , S_{\tau + n-1} - S_\tau \in dx_{n-1}, S_\tau = dy, \tau = m) \\ & = P(S_{m + n } \in dx+dy \vert S_{m + 1} \in dx_1 + dy, \dots , S_{m + n-1} \in dx_{n-1} + dy) \\ & = P(S_{m + n } \in dx+dy \vert S_{m + n-1} \in dx_{n-1} + dy) \\ & = P(S_{\tau + n } - S_\tau \in dx \vert S_{\tau + n-1} - S_\tau \in dx_{n-1}, S_\tau = dy, \tau = m) \\ & \overset{(1)}{=} P(S_{\tau + n } - S_\tau \in dx \vert S_{\tau + n-1} - S_\tau \in dx_{n-1}), \end{align} where I just used the definition of conditional probabilities when I wrote (1) over the equal sign. What do you think?