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There is a proof given here that I don't really understand, and was hoping someone more competent could explain it in some more detail:

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

In particular, how come we are guaranteed $X^\sim$ and $Y^\sim$ such that $X^\sim \sim X$ and $Y^\sim \sim Y$ with $X^\sim$ and $Y^\sim$ independent?

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For any set of distributions {D_1, ..., D_n}, we can always come up with a set of pairwise independent random variables {X_1, ..., X_n} s.th. X_i ~ D_i. This is why, without loss of generality, one can pick these two independent variables.

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