I have a question regarding the following theorem of the "two-series" :
Theorem : Let $(X_n)_{n \geq 1}$ be a sequence of independant and uniformely limited random variable. Then $\sum\limits_{n=1}^{+\infty} X_n$ converges $\iff$ $\sum\limits_{n=1}^{+\infty} E[X_n], \hspace{0.1cm}\sum\limits_{n=1}^{+\infty} \text{Var}[X_n]$ converge.
To imply that if $\sum\limits_{n=1}^{+\infty} X_n$ converge a.e then converge $\sum\limits_{n=1}^{+\infty} E[X_n]$ and $\sum\limits_{n=1}^{+\infty} \text{Var}[X_n]$, we costruct $(Y_n)_{n=1}^{+\infty}$ independant with same law and we afirm (this is the part I don't understand in which sense I'm supposed to think about it) that $\sum\limits_{n=1}^{+\infty}Y_n $ converges a.e, because the convergence is a property dependant only on the law of the sequence.
Details on this fact or clarification would be appreciated.