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De Finetti's theorem has already been mentioned, but it seems to me that it answers the original question. In this case, it says that any exchangeable infinite sequence $X_1$, $X_2$, $X_3$, ... X_1, X_2, X_3, \ldots$of real-valued random variables comes from some probability measure$\Phi$on the set of measures on$\Bbb R$. The sequence is generated by picking$\mu\sim\Phi$and then taking i.i.d.$X_1$,$X_2$,$X_3$, . ..$\sim\mu$. X_1, X_2, X_3, \ldots \sim\mu$. So, the third bullet point is automatically satisfied, and the "population distribution" is $\mu$. To get marginal normality you only need that ${\bf E}[\mu]=N(0,1)$, so there is a wide choice for $\Phi$.
De Finetti's theorem has already been mentioned, but it seems to me that it answers the original question. In this case, it says that any exchangeable infinite sequence $X_1$, $X_2$, $X_3$, ... of real-valued random variables comes from some probability measure $\Phi$ on the set of measures on $\Bbb R$. The sequence is generated by picking $\mu\sim\Phi$ and then taking i.i.d. $X_1$, $X_2$, $X_3$, ... $\sim\mu$. So, the third bullet point is automatically satisfied, and the "population distribution" is $\mu$. To get marginal normality you only need that ${\bf E}[\mu]=N(0,1)$, so there is a wide choice for $\Phi$.