I have a problem, let $S_n$ process such that $ S_n := \xi_1 + \dots + \xi_n$ and stoping time moment $\tau: \Omega \rightarrow \mathbb{N} \cup{\infty}$, where
\begin{equation*} \xi = \begin{cases} 1 & p \\ -1 & 1 - p \end{cases} \end{equation*}
I need to prove that $X_n = S_{n + \tau} - S_\tau$ has the same distribution like $S_n$ and independed of sigma algebra $\mathcal{F_\tau}$ and satisfies $X = S$ in distribution.
I know these facts are true for brownian motion by strong markov property, so I try to extend this theorem to this discrete case.
Can anyone tell me which direction I should go? I will appreciate any help.