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In Sinai's book on stochastic processes, the definition for discrete time stochastic processes is "a sequence of random variables $\{X_{n}\}_{n\in{}T}$ defined on a common probability space $(\Omega,\mathcal{B},P)$".

While I think it makes sense that the sequence of random variables may share a commen state space $(\Omega,\mathcal{B})$, I'm wondering why they also share the same probability measure? Aren't the random variables supposed to have different distributions as they move along time?

Can someone help explain where I am mistaken? Thank you!

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Careful. The distributions of the $X_n$ might be different or not. The distribution of $X_n$ is the probability measure $P\circ X_n^{-1}$. Now this is notationally elegant, but what it says is that for a Borel set $B\subseteq\mathbb{R}$, the distribution tells me how likely the value of $X_n$ lies in $B$. And that is $$P\big(\underbrace{\{\omega\in\Omega:X_n(\omega)\in B\}}_{X_n^{-1}}\big)=P\big(X_n^{-1}(B)\big).$$ So the distributions can be different even though the underlying probability space is the same. If you want to discuss things like whether $X_n$ and $X_{n+1}$ are independent, you have to go further and use their joint distribution.

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