Consider $\{S_n\}_{n\ge 0}$ to be the simple symmetric random walk with $S_0=0$ and $S_n=X_1+\cdots+X_n$ where $X_i$ are iid and $P(X_1=1)=P(X_1=-1)=1/2$. Is it possible to construct a process $\{Y_n\}_{n\in \mathbb{Z}}$ such that $\forall k\in \mathbb{Z}$ $$\{Y_n\}_{n\ge k}=\{S_n\}_{n\ge 1}$$
This was suggested to me by a teacher. I assume the equality of stochastic processes imply that all finite dimensional distributions are same. But I seem to have a discrepancy, though my teacher insists that's not possible. I'd appreciate if someone pointed out the mistake in my logic and pointed me towards the right direction.
So, assuming such a $\{Y_n\}_{n\in \mathbb{Z}}$ exists, we can take $k=1$ and then $Y_1$ and $S_1$ must have the same distribution. Hence $P(Y_1=\pm 1)=1$. But let's take $k=-2$ say. Then $Y_1$ and $S_4$ has the same distribution. But then $P(Y_1=4)=1/16>0$ contradicting $Y_1=\pm 1 \text{ ae}$. So no such stochastic process exists.