While studying the following theorem from Loeve's book on probability:
Let $X_n$ be uniformly bounded random variables. If $\sum X_n$ converges a.s. then $\sum \sigma^{2}X_n$ and $\sum E X_n$ converges
He writes the proof as follows:
To the random variables $X_n$ we associate random variables $X_n'$ such that $x_n$ and $X_n'$ are identically distributed for every n and $X_1,X_1',X_2,X_2'...$ is a sequence of independent random variables. We form the "symmetrized" sequence $X_{n}^{s} = X_n - X_n' $ of independent random variables.
So I checked the section on symmetrization but I cannot figure out the following:
-How can such a $X_n'$ be built in general? (they need to be independent and identically distributed) -How can I proof that the "symmetrized" sequence is made up with independent random variables?