This question is about a stationary process and the sigma-rings generated by its components.
Specifically, if $(X_{t})_{1}^{\infty}$ is a stationary process and $\mathscr{F}_{i}$ and $\mathscr{F}_{j}$ are the sigma-rings generated by $X_{i}$ and $X_{j}$ for $1 \leq i \neq j,$ then would we have $$\mathscr{F}_{i} = \mathscr{F}_{j}$$ for all $i \neq j$? Though intuitively this seems wrong, but I cannot see a reason.
Need your help, thanks!
No--that the process is stationary ensures that the distribution of $X_i$ does not depend on $i$, this has nothing to do with the identity $\sigma(X_i)=\sigma(X_j)$ for $i\ne j$, and in general these identities do not hold for stationary processes.
Example: if $(X_i)$ is i.i.d. Bernoulli, then $(X_i)$ is stationary but each $\sigma(X_i)$ is generated by $[X_i=1]$ and $[X_i=1]\notin\sigma(X_j)$ for every $i\ne j$.