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Hello I have a bit of an odd question. Can anyone think of an example of two time processes that have the same marginal distribution but differ in the joint distribution. So to be formal, two time processes $\{X_t\}_{t=1}^T$ and $\{Y_t\}_{t=1}^T$ st $F_t(X_t) = F_t(Y_t) \forall t$ but $F(X_1, ... X_T) \neq F(Y_1, ... Y_T)$

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  • $\begingroup$ Any two stationary Gaussian processes with different autocovariance functions (but the same mean function) will do. $\endgroup$
    – whuber
    Mar 3, 2020 at 17:14

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If you let $F = F_t \forall t$ and take $X_1 \sim F$ and $\ \forall t \neq 1; \ X_{t} = X_1 \sim F$, and simulate all $Y_t$ independantly from $ F$ , then indeed $F = F_t(X_t) = F_t(Y_t) \forall t$, and the joint distributions are $F(X_1,...X_T) = F \neq F(Y_1,...Y_T) = F^T$.

So basically you force all of the $X$s to be the same, and sample the $Y$s i.i.d

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