I was reading Amir Dembo's Probability Theory and there is a point that confuses me in the proof of Lindeberg's Central Limit Theorem for traingular arrays. He seems to use a result:
(Under some condition) if $X_1,X_2,...,X_n$ are multually independent random variables with finite mean $\{\mu_k\}$ and finite variance $\{\sigma_k^2\}$, then there exists random variables $Z_1,Z_2,...,Z_n$ with normal distributions $\{N(\mu_k,\sigma_k^2)\}$ on the same probability space such that $X_1,X_2,...,X_n,Z_1,Z_2,...,Z_n$ are mutually independent .$-------------(*)$
I think this statement, in general, is not true. But it may hold under some conditions. So I have a conjecture:
Conjecture: For a probability space $(\Omega,P)$, if the following condition is true: for every $n>0$ there exists random variables $Y_1,Y_2,...,Y_n$ that are mutually independent, then $(*)$ is correct.
Is this guess true? Or are there any other condition that makes $(*)$ correct?
Any help is appreciated!!